Department of Mathematics and Statistics, Binghamton University seminars:topsem

Fall 2014

2015-05-27T10:21:34-04:00

Fall 2014

September 11

Speaker: Ben McReynolds (Purdue University)

Title: Effective rigidity and counting

Abstract: In 1992, Alan Reid proved that if two arithmetic hyperbolic 2-manifolds have the same geodesic length spectrum, the two manifolds must be commensurable. In 2008, Chinburg-Hamilton-Long-Reid extended Reid's result to arithmetic hyperbolic 3-manifolds. In this talk, I will discuss effective versions of these results. Specifically, given two arithmetic hyperbolic 2- or 3-manifolds of some bounded volume V that are not commensurable, we ensure that a length L occurs in one but not both. More important, the length L can be bounded above as a function of the volume V and is explicitly given. These results rely on effective rigidity results for quaternion algebras. The main tools used are algebraic and geometric counting results of independent interest. Time permitting, I will discuss some of these counting results. This work is joint with Benjamin Linowitz, Paul Pollack, and Lola Thompson.

September 18

Speaker: Russell Ricks (University of Michigan)

Title: Flat strips in rank one CAT(0) spaces

Abstract: Let X be a proper, geodesically complete CAT(0) space under a geometric (that is, properly discontinuous, cocompact, and isometric) group action on X; further assume X admits a rank one axis. Using the Patterson-Sullivan measure on the boundary, we construct a generalized Bowen-Margulis measure on the space of geodesics in X. This additional structure allows us to prove some results about the original CAT(0) space X. Here are three such results: First, with respect to the Patterson-Sullivan measure, almost every point in the boundary of X is isolated in the Tit s metric. Second, under the Bowen-Margulis measure, almost no geodesic bounds a flat strip of any positive width. Third, we characterize when the length spectrum is arithmetic (that is, the set of translation lengths is contained in a discrete subgroup of the reals). In this talk, we will discuss the constructions and a few of the wrinkles involved for CAT(0) spaces.

September 25 no seminar.

October 2

Speaker: Zhiren Wang (Pennsylvania State University)

Title: Global Rigidity of Anosov $Z^r$ Actions on Tori and Nilmanifolds

Abstract: As part of a more general conjecture by Katok and Spatzier, it was asked if all smooth Anosov Z^r-actions on tori, nilmanifolds and infranilmanifolds without rank-1 factor actions are, up to smooth conjugacy, actions by automorphisms. In this talk, we will discuss a recent joint work with Federico Rodriguez Hertz that affirmatively answers this question.

October 9

Speaker: David Constantine (Wesleyan University)

Title: Some marked-length spectrum rigidity results for Fuchsian buildings

Abstract: I will present a few results on the question of marked-length spectrum rigidity for compact quotients of Fuchsian buildings. That is, I will detail some cases where knowing the length of the geodesic representative in each free homotopy class of loops is enough to recover the entire geometry of the building. Similar results hold for non-positively curved locally symmetric spaces and non-positively curved surfaces. This is joint work in progress with Jean-Francois Lafont.

October 16

Speaker: Andrew Nicol (Otterbein University)

Title: Quasi-isometries of graph manifolds do not preserve non-positive curvature

Abstract: In this talk, we will see the definition of high dimensional graph manifold and see that there are infinitely many examples coming from all dimensions 3 and higher of pairs graph manifolds with quasi-isometric fundamental groups, but where one supports a locally CAT(0) metric while the other cannot. We will use properties of the Euler class as well as various results on bounded cohomology.

October 23

October 30

Speaker: Matt Brin (Binghamton University)

Title: Groups of piecewise linear homeomorphisms of the unit interval

Abstract: It has long been a personal goal to understand the subgroup structure of the group of all piecewise linear, orientation preserving self homeomorphism of the unit interval, and it has been long accepted that this is not a particularly practical goal. However recent events have shown that while the ultimate goal might still be a long way off, we are a lot closer than we were a year ago. The talk is a progress report giving some results, some strong beliefs and some open questions. The work is joint with Collin Bleak and Justin Moore.

November 4 (cross listing with the Combinatorics Seminar)

Speaker: Emanuele Delucchi (Fribourg, Switzerland)

Title: Toric Arrangements – Towards Setting Up a Combinatorial Theory

Abstract: Recent work of De Concini, Procesi, and Vergne on vector partition functions gave a new impulse to the study of toric arrangements from algebraic, topological, and combinatorial points of view. In this context, many new discrete structures have appeared in the literature, each describing some aspect of the theory (i.e., either the arithmetic-algebraic one or the topological one) and trying to mirror the combinatorial framework which revolves around arrangements of hyperplanes. I will give a quick overview of the state of the art and, taking inspiration from some recent results of topological flavor, I will try to suggest a possible approach towards unifying these different objects.

November 13

Speaker: Matt Zaremsky (Binghamton University)

Title: HNN decompositions of Lodha-Moore groups, and topological applications

Abstract: In 1979, Geoghegan made four conjectures about Thompson's group F, three out of four of which have since been proved; the last one, non-amenability, is (in)famously still open. In 2013, Lodha and Moore found examples of finitely presented groups, closely related to F, which are non-amenable. They also showed that three out of four of Geoghegan's conjectures hold (this time including non-amenability). I have recently shown that they also satisfy the fourth conjecture, thus yielding the first examples of groups satisfying all four of Geoghegan's requirements. This talk will be an introduction to the Lodha-Moore groups, and a discussion of the key tool, that they are isomorphic to ascending HNN extensions of each other. If time permits I will also discuss my computation of the Bieri-Neumann-Strebel invariants for the Lodha-Moore groups.

November 20

Speaker: Christoforos Neofytidis (Binghamton University)

Title: Fundamental groups of aspherical manifolds and maps of non-zero degree

Abstract: We discuss obstructions to the existence of maps of non-zero degree from direct products to rationally essential manifolds, with special emphasis to aspherical manifolds whose fundamental groups have non-trivial center. As an application, we obtain an ordering of all non-hyperbolic 4-manifolds possessing a Thurston aspherical geometry.

November 27 no seminar.

December 4

December 11

Fall 2015

2015-12-15T13:34:38-04:00

Fall 2015

September 10
Speaker: Wiktor Mogilski (Binghamton University)
Title: The weighted Singer conjecture for Coxeter groups

Abstract:
Associated to a Coxeter system $(W,S)$ there is a contractible simplicial complex $\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly by reflections. Given an $S$-tuple of positive real numbers $\mathbf{q}$, one can define the weighted $L^2$-(co)homology groups of $\Sigma$ and associate to them a nonnegative real number called the weighted $L^2$-Betti number. Within the spectrum of weighted $L^2$-(co)homology there is a conjecture of interest called the Weighted Singer Conjecture, which was formulated in a 2007 paper of Davis–Dymara–Januszkiewicz–Okun. The conjecture claims that if $\Sigma$ is an $n$-manifold, then the weighted $L^2$-(co)homology groups of $\Sigma$ vanish above dimension $\frac{n}{2}$ whenever $\mathbf{q}\leq\mathbf{1}$ (that is, all terms of the multiparameter $\mathbf{q}$ are less than or equal to $1$). I will provide an overview of the current progress on the conjecture and present proofs of the conjecture (and variations of it) when $n=3,4$..
September 17
Speaker: Dmytro Yeroshkin (Syracuse University)
Title: On Poincaré Duality for Orbifolds

Abstract:
This talk will examine the obstructions to integer-valued Poincaré duality for (underlying spaces of) orbifolds. In particular, it will be shown that in dimensions 4 and 5, the obstruction is controlled by the orbifold fundamental group. A consequence of this is that if the orbifold fundamental group is naturally isomorphic to the fundamental group of the underlying space, then the orbifold satisfies integer-valued Poincaré duality.
September 24
Speaker: Federico Rodriguez Hertz (Pennsylvania State University)
Title: Stable ergodicity in low dimensions

Abstract:
We shall give an overview of what is known about stable ergodicity for partially hyperbolic systems in low dimensions (either of the manifold or of the center bundle). We shall also state some old and new problems and give the main obstacles towards their solution.
October 1 Joint with Combinatorics Seminar
Speaker: Florian Frick (Cornell University)
Title: Equivariant topology of mass partitions by hyperplanes

Abstract:
I will review the configuration-space/test-map scheme which is central to the study of many problems in topological combinatorics. I will then apply this method to investigate the existence of fair partitions of measures in Euclidean space by affine hyperplanes. One important instance of this is the Ham Sandwich theorem, which guarantees that any sandwich made of bread, ham, and cheese can be fairly cut into two pieces with a long knife. I will use equivariant obstruction theory to prove results about higher-dimensional sandwiches (that is, collections of measures) that have to be cut into several equal pieces.

This is joint work with Pavle Blagojevic, Albert Haase, and Gunter M. Ziegler.
October 8

Dean's Lecture in Geometry and Topology.
Speaker: Christopher Croke (University of Pennsylvania)
Title: Geometric Rigidity Problems

Abstract:
In this talk we introduce some geometric rigidity problems that ask if you can determine a compact Riemannian manifold with boundary from measurements taken from the outside. The problems includes the boundary rigidity problem: Can you determine the metric inside if you know the distances (measured through the inside) between all pairs of boundary points? The lens rigidity problem: Can you determine the metric if you know how each entering geodesic exits and how long it takes to exit? We will present some counterexamples and some theorems as well as mention some relations to other problems and potential applications.

October 13, 2:50pm Joint with algebra seminar.
Speaker: Robert Bieri (Binghamton University)
Title: Groups of piecewise isometric permutations of lattice points

Abstract:
An orthant (of the orthogonal integral lattice) $L\subseteq \mathbb Z^n$ is the image of the standard orthant $\mathbb ℕ^n$ under an affine-orthogonal transformation. And a permutation $p: S\longrightarrow S$ of a subset $S\subseteq \mathbb Z^n$ is piecewise-Euclidean-isometric (pei), if $S$ is a disjoint union of finitely many orthants, $S = \bigcup_{i}L_i$, on each of which $p$ restricts to an isometric embedding $L_i\rightarrow S$. I will talk about the group $\text{pei}(S)$ of all pei-permutations of various subsets $S$ and its subgroup $\text{pet}(S)$ of all piecewise-Eucliden-translation permutations. In the case when $S$ consists of the lattice points on the union of $n$ positive coordinate axes then $\text{pet}(S)$ is the Houghton group $H_n$.

This is joint work with Heike Sach: we prove finiteness properties of some pei- and pet- groups, and have, as a consequence, that $\text{pei}(\mathbb Z^n)$ admits a K(G,1)-complex with finite $(2^n – 1)$-skeleton.

An interesting point is that prominent groups like Richard Thompson's group $V$ crop up when we extend the consideration to the $\text{SL}_2(\mathbb Z)$-lattice in the hyperbolic plane..

October 15
Speaker: Andrew Geng (University of Chicago)
Title: Classification and examples of 5-dimensional geometries

Abstract:
Thurston's eight homogeneous geometries formed the building blocks of 3-manifolds in the Geometrization Conjecture. Filipkiewicz classified the 4-dimensional geometries in 1983, finding 18 and one countably infinite family. I have recently classified the 5-dimensional geometries. I will review what a geometry in the sense of Thurston is, survey related ideas, and outline the classification in 5 dimensions. Salient features, especially those first occurring in dimension 5, will be illustrated using particular geometries from the list. The classification touches a number of topics including foliations, fiber bundles, representations of compact Lie groups, Lie algebra cohomology, Galois theory in algebraic number fields, and conformal transformation groups. I hope to give some indication of how all of these come into play..
October 22 Joint with Analysis Seminar
Speaker: Jiuyi Zhu (John Hopkins University)
Title: Doubling estimates, vanishing order and nodal sets of Steklov eigenfunctions

Abstract:
Recently the study of Steklov eigenfunctions has been attracting much attention. We investigate the qualitative and quantitative properties of Steklov eigenfunctions. We obtain the sharp doubling estimates for Steklov eigenfunctions on the boundary and interior of the manifold using Carleman inequalities. As an application, optimal vanishing order is derived, which describes quantitative behavior of strong unique continuation property. We can ask Yau's type conjecture for the Hausdorff measure of nodal sets of Steklov eigenfunctions. We derive the lower bounds for interior and boundary nodal sets. In two dimensions, we are able to obtain the upper bounds for singular sets and nodal sets. Part of work is joint with Chris Sogge and X. Wang.
October 29
Speaker: Anisah Nu'Man (Trinity College)
Title: Intrinsic tame filling functions

Abstract:
Let $G$ be a group with a finite presentation $P = \langle A|R \rangle$ such that $A$ is inverse-closed. Let $f \colon \mathbb{N}[\frac{1}{4}] \rightarrow \mathbb{N}[\frac{1}{4}]$ be a nondecreasing function. Loosely, $f$ is an intrinsic tame filling function for $(G,\mathcal{P})$ if for every word $w$ over $A^∗$ that represents the identity element in $G$, there exists a van Kampen diagram $\triangle$ for $w$ over $P$ and a continuous choice of paths from the basepoint ∗ of $\triangle$ to the boundary of $\triangle$ such that the paths are steadily moving outward as measured by $f$. The isodiametric function (or intrinsic diameter function) introduced by Gersten and the extrinsic diameter function introduced by Bridson and Riley are useful invariants capturing the topology of the Cayley complex. Tame filling functions are a refinement of the diameter functions introduced by Brittenham and Hermiller and are used to gain insight on how wildly maximum distances can occur in van Kampen diagrams. Brittenham and Hermiller showed that tame filling functions are a quasi-isometry invariant and that if $f$ is an intrinsic (respectively extrinsic) tame filling function for $(G,\mathcal{P})$, then $(G,\mathcal{P})$ has an intrinsic (respectively extrinsic) diameter function equivalent to the function $n \rightarrow [f(n)]$. In contrast to diameter functions, it is unknown if every pair $(G,\mathcal{P})$ has a finite-valued tame filling function. In this talk I will discuss intrinsic tame filling functions for graph products (a generalization of direct and free products) and certain free products with amalgamation.

November 9, 3:30pm Note special day and time!
Speaker: Renato G. Bettiol (University of Pennsylvania)
Title: Positive biorthogonal curvature in dimension 4

Abstract:
A 4-manifold is said to have positive biorthogonal curvature if the average of sectional curvatures of any pair of orthogonal planes is positive. In this talk, I will describe a construction of metrics with positive biorthogonal curvature on the product of spheres, and then combine it with recent surgery stability results of Hoelzel to classify (up to homeomorphism) the closed simply-connected 4-manifolds that admit a metric with positive biorthogonal curvature.
November 19
Speaker:
Title: TBA

Abstract:
TBA.
December 3
Speaker: Matt Brin (Binghamton)
Title: Elementary amenable subgroups of PL homeomorphisms of the interval that are not very elementary.

Abstract:
Among the piecewise linear homeomorphisms of the unit interval, we find a sequence of easily described 2-generator groups of rapidly increasing complexity. In spite of their complexity, we can (oxymoronically) calculate an exact measure of each group's complexity. We will define terms, give the exact contructions and give some details of the analysis..

Fall 2016

2017-01-12T08:59:47-04:00

Fall 2016

For questions contact Christoforos Neofytidis

September 8
Speaker: Phillip Wesolek (SUNY at Binghamton)
Title: Elementary amenable groups and the space of marked groups

Abstract: (joint work with J. Williams.) The space of marked groups is a compact totally disconnected space that parameterizes all countable groups. This space allows for tools from descriptive set theory to be applied to study group-theoretic questions. In this talk, we consider the collection of elementary amenable marked groups. The class of elementary amenable groups is the smallest class that contains the abelian groups and the finite groups and that is closed under group extension, taking subgroups, taking quotients, and taking countable directed unions. We give a characterization of elementary amenable groups in terms of a chain condition. We then show the set of elementary amenable marked groups is not in the Borel sigma algebra of the space of marked groups. This gives a new, non-constructive proof of a theorem of Grigorchuk: There are finitely generated amenable non-elementary amenable groups.

September 29
Speaker: Wiktor Mogilski (SUNY at Binghamton)
Title: L^2-cohomology: Conjectures Abound!

Abstract: I will present a brief introduction to L^2-cohomology and then discuss some conjectures lurking amidst us, as well as their implications. I will then survey some recent results and developments.

October 6
Speaker: Wiktor Mogilski (SUNY at Binghamton)
Title: L^2-cohomology: Conjectures Abound! (Part II)

October 13
Speaker: Steve Ferry (SUNY at Binghamton and Rutgers University)
Title: Distance functions, data, and comparison geometry

Abstract: We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function, an observation that has consequences in topological data analysis.

October 20
Speaker: Thomas Barthelmé (Queen's University)
Title: Counting orbits of Anosov flows in free homotopy classes

Abstract: (Joint work with Sergio Fenley) Since Margulis and Bowen gave an estimate of the growth rate of periodic orbits of Anosov flow, there has been a lot of research furthering counting questions. If one consider only Anosov flows, these developments have been either into giving more precise estimates or into counting periodic orbits given a homological constraint, i.e., counting periodic orbits that are in the same fixed homology class. I will talk here about a third direction: Despite what one might think when considering the most classical examples of Anosov flows, a lot of Anosov flows (maybe most) in 3-manifolds are such that some periodic orbits are freely homotopic to infinitely many other. It is therefore legitimate to ask whether one can give an estimate of the growth rate of periodic orbits inside an infinite free homotopy class. I will explain how one can use the geometry and topology of Anosov flows in 3-manifolds to obtain such estimates. As a corollary, we get an answer to the following question, asked by Plante and Thurston in 1972: If M is a manifold supporting an Anosov flow, does the number of conjugacy classes in the fundamental group grows exponentially fast with the length of the shortest orbit representative?

October 27
Dean's Lecture in Geometry and Topology.
Speaker: Boris Hasselblatt (Tufts University)
Title: Statistical properties of deterministic systems by elementary means

Abstract: The Maxwell-Boltzmann ergodic hypothesis aimed to lay a foundation under statistical mechanics, which is at a microscopic scale a deterministic system. Similar complexity was discovered by Poincaré in celestial mechanics and by Hadamard in the motion of a free particle in a negatively curved space. We start with a guided tour of the history of the subject from various perspectives and then discuss the central mechanism that produces pseudorandom behavior in these deterministic systems, the Hop argument. It has been known to extend well beyond the scope of its initial application in 1939, and we show that it also leads to much stronger conclusions: Not only do time averages of observables coincide with space averages (which was the purpose for making the ergodic hypothesis), but any finite number of observables will become decorrelated with time. That is, the Hopf argument does not only yield ergodicity but mixing, and often mixing of all orders.

November 3
Speaker: William Menasco (SUNY at Buffalo)
Title: Efficient geodesics and an effective algorithm for distance in the complex of curves

Abstract: This talk will present joint work with Dan Margalit and Joan Birman, an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, this approach is new, simple, and more effective for all distances accessible by computer. The method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky.

November 3
Colloquium (4:30-5:30 pm): William Menasco (SUNY at Buffalo)

November 10
Speaker: Bena Tshishiku (Harvard University)
Title: Obstructions to Nielsen realization

Abstract: Let M be a manifold, and let Mod(M) be its mapping class group. The Nielsen realization problem for diffeomorphisms asks, “Can a given subgroup G<Mod(M) be lifted to the diffeomorphism group Diff(M)?” This question about group actions is related to a question about flat connections on fiber bundles with fiber M. In the case M is a closed surface, the answer is “yes” for finite G (by work of Kerckhoff) and “no” for G=Mod(M) (by work of Morita). For most infinite G<Mod(M), we have no idea. I will discuss some obstructions that can be used to show that certain groups don’t lift. Some of this work is joint with Nick Salter.

November 17 (Joint with Combinatorics Seminar)
Note special time and room: Time: 1:15 - 2:15, Room: WH-329
Speaker: Eric Babson (U. C. Davis)
Title: Gaussian Random Knots

Abstract: A model for random knots or links is obtained by fixing an initial curve in some n-dimensional Euclidean space and projecting the curve to random three dimensional subspaces. By varying the curve we obtain different models of random links. I will study how the second moment of the average linking numbers change as a function of the initial curve. This is based on work of Christopher Westenberger.

November 17
Speaker: Todd Fisher (Brigham Young University)
Title: Unique equilibrium states for geodesic flows in nonpositive curvature

Abstract:The geodesic flow for a compact Riemannian manifold with negative curvature has a unique equilibrium state for every Holder continuous potential function. This is no longer true if the curvature is only nonpositive. We show that there is a large class of potentials with unique equilibrium states. Specifically, we prove that for compact rank 1 surfaces of nonpositive curvature that the a scalar times geometric potential has a unique equilibrium state for the scalar less than 1. Furthermore, if a potential satisfies a bounded range hypothesis for compact rank 1 manifolds with nonpositive curvature, then there will be a unique equilibrium state. This is joint work with Keith Burns, Vaughn Climenhaga, and Dan Thompson.

Spring 2015

2015-05-27T10:19:54-04:00

Spring 2015

February 5

Speaker: Ivan Izmestiev (Freie Universität Berlin)

Title: Variational properties of the discrete Hilbert-Einstein functional

Abstract: The discrete Hilbert-Einstein functional (also known as Regge action) for a 3-manifold glued from euclidean simplices is the sum of edge lengths multiplied with angular defects at the edges. There is an analog for hyperbolic cone-manifolds; a discrete total mean curvature term appears if the manifold has a non-empty boundary. Variational properties of this functional are similar to those of its smooth counterpart. In particular, critical points correspond to vanishing angular defects, i.e. to metrics of constant curvature. We give a survey on isometric embeddings and rigidity results that can be obtained by studying the second derivative of the discrete Hilbert-Einstein and speak about possible future developments.

February 12

Speaker: James Dibble (Rutgers University)

Title: Totally geodesic maps into manifolds with no focal points

Abstract: A classical result of Eells-Sampson is that every homotopy class of maps between compact Riemannian manifolds, where the target has non-positive sectional curvature, contains an energy-minimizing harmonic representative. They proved this by inventing the harmonic map heat flow, the first geometric flow defined on manifolds. Their work was refined by Hartman, who proved the monotonicity of certain distance functions under the flow and used this to deduce that that the space of harmonic maps in each homotopy class is path-connected and that energy is constant on it. Applying an identity that dates to the work of Bochner, Eells-Sampson also proved that, when the domain has non-negative Ricci curvature, all harmonic maps are totally geodesic.

It will be shown that, for domains with non-negative Ricci curvature, the results of Eells-Sampson, along with certain qualitative consequences of Hartman's results, generalize to energy-minimizing maps into manifolds with no focal points. These are manifolds whose universal covers satisfy a simple synthetic condition: For each point and each maximal geodesic, there is a unique geodesic connecting them that intersects the latter perpendicularly. By contrast with previous approaches, the proof uses neither a geometric flow nor the Bochner identity for harmonic maps.

February 19

Colloquium at 4:30pm

Speaker: Jonathan Williams (University of Georgia)

Title: A new approach to general smooth 4-manifolds

Abstract: Some consider smooth 4-manifolds to be a mature field, which typically means its approachable yet nontrivial problems have become scarce. This is mainly due to a lack of tools. In this talk I will present a new way to depict any smooth, closed oriented 4-manifold that opens the doors to two of the most successful tools from 3-manifolds: pseudoholomorphic curves and discrete groups.

February 26

Speaker: Eric Swartz (Western Australia)

Title: Generalized quadrangles with symmetry

Abstract: A generalized quadrangle is a point-line incidence geometry Q such that (1) any two points lie on at most one line, and (2) given a line l and a point P not incident with l, P is collinear with a unique point of l. Generalized quadrangles are a specific type of generalized polygon, which were first introduced by Tit s in 1959 as geometries associated to classical groups. It is natural, then, to ask the question: if one starts with the abstract definition of a generalized quadrangle, which ones are highly symmetric? I will discuss the background of this question, leading to the following recent work:

An antiflag of a generalized quadrangle is a non-incident point-line pair (P, l), and we say that the generalized quadrangle Q is antiflag-transitive if the group of collineations (automorphisms that send points to points and lines to lines) is transitive on the set of all antiflags. We prove that if a finite, thick generalized quadrangle Q is antiflag-transitive, then Q is one of the following: the unique generalized quadrangle of order (3,5), a classical generalized quadrangle, or a dual of one of these.

This is joint work with John Bamberg and Cai-Heng Li, and this talk will assume no prior knowledge of finite geometry.

Colloquium at 4:30pm

Speaker: Niels Martin Moeller (Princeton University)

Title: Gluing of Geometric PDEs - Obstructions vs. Constructions for Minimal Surfaces & Mean Curvature Flow Solitons

Abstract: For geometric nonlinear PDEs, where no easy superposition principle holds, examples of (global, geometrically/topologically interesting) solutions can be hard to come about. In certain situations, for example for 2-surfaces satisfying an equation of mean curvature type, one can generally “fuse” two or more such surfaces satisfying the PDE, as long as certain global obstructions are respected - at the cost (or benefit) of increasing the genus significantly. The key to success in such a gluing procedure is to understand the obstructions from a more local perspective, and to allow sufficiently large geometric deformations to take place. In the talk I will introduce some of the basic ideas and techniques (and pictures) in the gluing of minimal 2-surfaces in a 3-manifold. Then I will explain two recent applications, one to the study of solitons with genus in the singularity theory for mean curvature flow (rigorous construction of Ilmanen's conjectured “planosphere” self-shrinkers), and another to the non-compactness of moduli spaces of finite total curvature minimal surfaces (a problem posed by Ros & Hoffman-Meeks). Some of this work is joint w/ Steve Kleene and/or Nicos Kapouleas.

March 5

Speaker: Ross Geoghegan (Binghamton University)

Title: A theorem about extensions of groups

March 12

March 19

Speaker: Matt Zaremsky (Binghamton University)

Title:The $\Sigma$-invariants of Thompson's group $F$, via Morse theory

Abstract: In a paper published in 2010, Bieri, Geoghegan and Kochloukova computed the $\Sigma$-invariants (also called Bieri-Neumann-Strebel-Renz invariants) of Thompson's group $F$. In recent joint work with Stefan Witzel, we recomputed these using the action of $F$ on a certain CAT(0) cube complex called the Stein-Farley complex. The main tool is a version of discrete Morse theory. I will explain what all of these words mean over the course of the talk, and it should be accessible to non-experts.

March 26

Speaker: Matt Zaremsky (Binghamton University)

Title: The $\Sigma$-invariants of the generalized Thompson's groups $F_n$

Abstract: Building off the first talk, I will shift from $F$ to a family of groups $F_n$, of which $F$ is $F_2$. Using the action of $F_n$ on a CAT(0) cube complex, I was recently able to compute all the $\Sigma$-invariants of all the $F_n$. In this talk I will focus on the aspects of the $\Sigma$-invariants that only come up when $n>2$, and will highlight a new technique, building off work of Belk and Forrest, for proving higher connectivity properties of certain complexes. This talk will still be accessible to non-experts, though it will help to have gone to the first talk.

April 2

Speaker: Jim Belk (Bard College)

Title: Rearrangement Groups of Fractals

Abstract: The definition of Thompson's group $F$ depends crucially on the self-similar structure of the unit interval. In this talk, I will describe a family of Thompson-like groups that act on a variety of self-similar structures. Each of these groups has an associated CAT(0) cubical complex, analogous to the Farley complexes for $F$, $T$, and $V$. By analyzing descending links on these complexes, I will show that some of these groups have type $F_\infty$. This is joint work with Bradley Forrest.

April 9 no seminar.

April 16

Speaker: Stefan Witzel (Bielefeld University)

Title: Arithmetic groups, finiteness properties, and homology

Abstract: The group $SL_2(F_p[t,t^{-1}])$ is finitely generated but not finitely presented. In fact, it has a finite-index subgroup $G$ with $H_2(G,F_p)$ infinite (this and more was shown by Stuhler). I will talk about results of the same kind for related groups.

April 22 at 3:30pm in WH 309

Speaker: Ralf Spatzier (University of Michigan)

Title: Higher Rank Rigidity and Positive Curvature

Abstract: I will review rigidity and non-rigidity results about “higher rank” in Riemannian geometry. Specifically we consider “higher rank” spaces in which subobjects of extremal curvature are plentiful. I will emphasize recent joint work with Schmidt and Shankar on Riemannian manifolds of higher spherical rank where every geodesic c has a perpendicular parallel field making sectional curvature 1 with c, and the sectional curvature is bounded below by 1.

April 23

Hilton Memorial Lecture at 3pm in Science II, Room 140.

Speaker: Ralf Spatzier (University of Michigan)

Title: Higher Rank in Geometry and Dynamics - How isometric and hyperbolic behavior force rigidity

Abstract: Higher rank phenomena have led to surprising rigidity results in group theory, geometry and dynamics. Examples start with Margulis superrigidity theorem for lattices in higher rank semisimple Lie groups, followed by the classification of nonpositively curved Riemannian manifolds with lots of flats. In recent years similar phenomena have been found in dynamics, in particular in the classification of hyperbolic actions on tori and nilmanifolds of higher rank Abelian groups and their measure rigidity.

April 30

Dean's Lecture in Geometry and Topology.
Speaker: Karsten Grove (Notre Dame University)

Title: Symmetry and Positive Curvature

Abstract: Although constituting a vast extension of ancient Spherical Geometry, the beautiful class of positively curved (Riemannian) spaces is like the “Tip of the Iceberg” among all (Riemannian) spaces. Accordingly, non-symmetric positively curved spaces are known only in a few sporadic dimensions, and yet only a few obstructions to their existence are known.

In this talk, we will describe the current state of affair of the subject including tools and methods, with emphasis on the impact symmetries have had on the development during the last few decades.

May 7

Speaker: Boris Kalinin (Pennsylvania State University)

Title: Smooth rigidity and classification for hyperbolic systems and actions

Abstract: Hyperbolic actions of $\mathbb{Z}^k$ and $\mathbb{R}^k$ extend the classical notion of Anosov diffeomorphisms and flows, which are hyperbolic actions of $\mathbb{Z}$ and $\mathbb{R}$. In contrast to the rank one case, higher rank hyperbolic actions exhibit various rigidity properties. I will focus on the problem of smooth classification, that is finding a smooth conjugacy to an algebraic model. I will give an overview of this area and compare it with the rank one case, where the natural problem is either topological classification or smooth classification under extra assumptions.

Spring 2016

2016-07-15T14:21:55-04:00

Spring 2016

February 11
Speaker: Peng Sun (Central University of Finance and Economics, Beijing, China)
Title: Exponential Decays and Topological Entropy

Abstract: We study two types of exponential decay under dynamics: Lebesgue number of open covers and expansive constants for expansive maps. We find that these exponents can provide better estimates of topological entropy.
February 18
Speaker: Artem Dudko (SUNY Stony Brook)
Title: On representations of weakly branch groups.

Abstract: The class of weakly branch groups acting on rooted trees plays important role in group theory and dynamics and contains many examples of groups with unusual properties. I'll present results on representations associated to actions of weakly branch groups on boundaries of rooted trees and corollaries related to invariant random subgroups, centralizers of group actions, spectra of Schreier graphs etc. The talk is based on a joint work with R. Grigorchuk.

March 3
Speaker: Steven Frankel (IAS)
Title: Quasigeodesic and pseudo-Anosov flows

Abstract: We will discuss two kinds of flows on 3-manifolds: quasigeodesic and pseudo-Anosov. Quasigeodesic flows are defined by a tangent condition, that each flowline is coarsely comparable to a geodesic. In contrast, pseudo-Anosov flows are defined by a transverse condition, where the flow contracts and expands the manifold in different directions.

When the ambient manifold is hyperbolic, there is a surprising relationship between these apparently disparate classes of flows. We will show that a quasigeodesic flow on a closed hyperbolic 3-manifold has a “coarsely contracting-expanding” transverse structure, and use this to show that every such flow has closed orbits. We will also illustrate an approach to Calegari’s conjecture, that every quasigeodesic flow can be deformed into a pseudo-Anosov flow.
March 10
Speaker: Andrey Gogolev (Binghamton)
Title: Surgery constructions of Anosov flows

Abstract: I will recall classical Franks-Williams construction of a non-transitive Anosov flow. Then I will explain how Franks-Williams idea can be transplanted into the setting of geodesic flows.
The first half of the talk will be dedicated to dynamical preliminaries and will be more entertaining than normal.
March 17
Speaker: Jean Lafont (Ohio State university)
Title: Almost-isometries: rigidity versus flexibility.

Abstract: Almost-isometries are quasi-isometries with multiplicative constant 1, i.e. maps which distort distances by some uniform additive amount. Given a pair of Riemannian metrics on a closed manifold, we can look at their lifts to the universal cover, and ask whether or not these metrics are almost-isometric (it is easy to see they are quasi-isometric). In the rigidity direction, there are many cases where the only times these metrics are almost-isometric is if they are actually isometric (joint with Kar and Schmidt). In the flexibility direction, we give examples where there is an infinite dimensional space of metrics whose lifts are almost-isometric, but where no two are isometric (joint with Schmidt and van Limbeek). The talk will be accessible to a general audience.

April 7
Speaker: Tam Nguyen Phan (Binghamton University)
Title: Finite volume, noncompact manifolds of negative curvature

Abstract:Let M be a noncompact, complete, Riemannian manifold. Gromov proved that if the sectional curvature of M is negative and bounded, and if the volume of M is finite, then M is homeomorphic to the interior of a compact manifold with boundary. In other words, M has finitely many ends, and each end of M is topologically a product $C\times [0,\infty)$ of a closed manifold $C$ with a ray. I will discuss the question what topological restrictions there are on each end of such a manifold M. The talk will be accessible to a general audience.
April 14
Speaker: Matt Zaremsky (Binghamton University)
Title: Finiteness properties of some subgroups of the pure braid groups

Abstract:The Bieri-Neumann-Strebel-Renz invariants of a group are a sequence of geometric objects that encode a great deal of information about certain subgroups of the group, including “finiteness properties” like finite generation and finite presentability. In general they are quite difficult to compute, and a full computation has been done only for very few “interesting” families of groups. I will discuss some of my results on the BNSR-invariants of the pure braid groups, and the implications for finiteness properties of their subgroups. In particular I will discuss some natural subgroups that are finitely generated but not finitely presented, finitely presented but not of type F_3, type F_3 but not F_4, and so forth.
April 21
Speaker: Christoforos Neofytidis (Binghamton)
Title: Mapping degree sets of Cartesian products

Abstract:We study stability of properties of closed oriented manifolds under taking non-trivial Cartesian products, with special emphasis to properties related to the sets of self-mapping degrees. We derive applications with respect to the non-existence of orientation reversing self-maps.

April 28 No seminar due to a special event:
Peter Hilton Memorial Lecture.
Speaker: Amie Wilkinson (University of Chicago)
Title: Geometry, Lyapunov Exponents and Rigidity

May 5
Speaker: Kevin Schreve (University of Michigan)
Title: Thurston norm via Fox Calculus

Abstract:Given a 3-manifold whose fundamental group admits a presentation with two generators and one relator we will show how one can easily determine the Thurston norm. This is based on joint work with Stefan Friedl and Stephan Tillmann.
May 9 (Working seminar, 4:40-5:40, WH-100E)
Speaker: Barry Minemyer (Ohio State University)
Title: The isometric embedding problem and Nash's $C^1$ solution

May 11 (Working seminar, 3:30-4:30, WH-309)
Speaker: Barry Minemyer (Ohio State University)
Title: The Nash smooth isometric embedding theorem

May 12
Speaker: Barry Minemyer (Ohio State University)
Title: The isometric embedding problem for length metric spaces

Abstract:In the spring of 2010, Pedro Ontaneda suggested to me the research question of whether or not the famous Nash isometric embedding theorems could be extended to geodesic metric spaces. In this talk I will begin by explaining the general problem, discuss what is known in the more concrete cases of manifolds and polyhedra, and end with a new result which applies to a much larger class of spaces than anything that was previously known.

Fall 2017

2019-02-18T08:50:30-04:00

Fall 2017

July 17
Speaker: Collin Bleak (University of St. Andrews)
Title: On Finite generation for groups of homeomorphisms of Cantor spaces

Abstract: Given a Cantor space $c$, we define a broad class of subgroups of the group of homeomorphisms of $c$ so that, if a given group $G$ is a subgroup of a finitely generated subgroup $H$ of $Homeo(c)$, and is in our class, then we can immediately conclude that $G$ is two-generated. The argument has connections with Higman and Epstein's general arguments toward simplicity for groups of homeomorphisms, and is general enough to immediately prove two-generation, e.g., for many relatives of the Thompson groups, and many other groups as well. Joint with James Hyde.

August 24

Organizational meeting

August 31 (Joint with combinatorics)
Speaker: Olakunle Abawonse (Binghamton University)
Title: Topological Tverberg Theorem (prime power case)

Abstract: We will solve some discrete geometry problems using methods of equivariant topology. This talk is based on the paper “Beyond The Borsuk-Ulam Theorem – The Topological Tverberg Story” by Blagojevic and Ziegler. Topological techniques ranging from the Borsuk-Ulam theorem to spectral sequences will be used in solving these problems.

September 7
Speaker: Casey Donoven (Binghamton University)
Title: Topology of Fractals

Abstract: Invariant factors are quotients of Cantor space that generalize the topology of certain fractals. They provide insight into the topology of self-similar sets and Julia sets and are interesting in their own right. Under certain conditions, invariant factors are inverse limits of finite topological spaces realizable as finite graphs. In this talk, I will present basic definitions and results pertaining to invariant factors and motivate them through poorly-drawn examples of self-similar fractals.

September 14
Speaker: Jun Zhang (Tel Aviv University)
Title: Applications of persistent homology in symplectic and Riemannian geometry

Abstract: In this talk, I will start from a brief introduction of persistent homology and its related formulations from the perspective of symplectic geometry, especially in various forms of Floer theory. Then I will quickly demonstrate via several examples how persistent homology is used in symplectic geometry, for instance in solving some (Hamiltonian) dynamical problems. Finally, I will focus on a recent work (joint with V. Stojisavljevic) on an application in Riemannian geometry - quantitatively comparing Riemannian metrics, which is based on a newly developed concept called symplectic Banach-Mazur distance.

September 21 (no seminar - Rosh Hashahah)

September 28
Speaker: Oleg Lazarev (Columbia University)
Title: Contact manifolds with flexible fillings

Abstract: In this talk, I will show that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology. As an application, in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many exotic contact structures. Using similar methods, I will also construct an infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These results are proven by studying Reeb chords of loose Legendrians and positive symplectic homology.

October 5
Speaker: Michael Cohen (North Dakota State University)
Title: Polishability of some groups of interval and circle diffeomorphisms

Abstract: Consider a group $G$ consisting of all $C^k$ diffeomorphisms of the circle whose derivatives satisfy a regularity condition arising from classical real analysis: a Lipschitz/Hoelder condition; absolute continuity; or bounded total variation. Is it possible to assign a separable complete metric topology to $G$, in such a way that the group operations become continuous? If so, $G$ is called Polishable. I'll discuss this Polishability problem in the cases mentioned above, where the answer turns out to vary dramatically depending on the choice of analytic condition. In particular, I'll exhibit an infinite class of what appear to be new Polish topological groups.

October 12
Speaker: Robert Kropholler (Tufts University)
Title: Uncountably many QI classes of groups of type FP

Abstract: The first groups of type FP which were not of type F were discovered by Bestvina and Brady in 1997. More recently Leary has shown that there are uncountably many groups of type FP.

In 1998 Bowditch showed that there uncountably many QI classes of 2-generator groups.

In joint work with Ian Leary and Ignat Soroko, we combine these ideas and prove there are uncountably many QI classes of groups of type FP. I will discuss the previous work of Bestvina, Brady, Bowditch and Leary leading up to this project and then talk about how these elements come together.

October 19
Speaker: Ramón Vera (Institute of Mathematics, National Autonomous University of Mexico)
Title: Poisson Structures of near-symplectic Manifolds

Abstract: In this talk we will show a connection between two singular geometric structures: near-symplectic forms and Poisson structures. Near-symplectic forms were introduced by Taubes as a way of generalizing symplectic topology in dimension 4. These structures are closely related to broken Lefschetz fibrations, which can be seen as extensions of Lefschetz pencils. We will describe some aspects of near-symplectic manifolds in any dimension 2n. On the other hand, Poisson structures have their origin in Classical Mechanics. A Poisson bivector naturally determines a singular foliation by symplectic leaves. We will discuss the link between these geometries and some features of their Poisson cohomology. This is joint work with P. Batakidis.

October 26
Speaker: Adam Saltz (University of Georgia)
Pictorial link homology (and Floer homology?): TBA

Abstract: First, I'll describe Bar-Natan's amazing reformulation of Khovanov homology using diagrams and cobordisms (and a bit of category theory) rather than linear algebra. This will be totally accessible to graduate students!

Then I'll tell you about a strategy to show that link cobordisms induce well-defined maps on “Khovanov-Floer theories” (Floer homology theories which admit spectral sequences from Khovanov homology). This strengthens a result of Baldwin, Hedden, and Lobb. I also think it's a good start on parametrizing these theories, and I'll tell you how.

November 2
Speaker: Eugenia Sapir (Binghamton University)
Title: Long geodesics on surfaces

Abstract: I will talk about a recent result of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of this tessellation. Along the way, we look at how long geodesic arcs behave in very small balls on the surface.TBA

November 9
Speaker: Cary Malkiewich (Binghamton University)
Title: Periodic points and equivariant stable homotopy theory

Abstract: The Lefschetz number $L(f)$ and Reidemeister trace $R(f)$ are invariants that detect fixed points of a map $f: X \to X$. I will talk about the generalizations of these invariants that detect n-periodic points of $f$, and an ongoing project with Kate Ponto that involves showing that periodic-point problems and equivariant fixed-point problems are controlled by the same invariants. Along the way we will see some equivariant stable homotopy theory, and some arguments that amount to ``unwinding'' a string to show that a more complicated invariant reduces to a simpler one. These arguments allow us to resolve a conjecture of Klein and Williams; it still remains to connect them to related work of Geoghegan and Nicas on flows.

November 16
Speaker: Mark Sapir (Vanderbilt University)
Title: Divergence functions of R. Thompson groups

Abstract: This is a joint work with Gili Golan. The divergence function of a group generated by a finite set $X$ is the smallest function $f(n)$ such that for every $n$ every two elements of length $n$ can be connected in the Cayley graph (corresponding to $X$) by a path of length at most $f(n)$ avoiding the ball of radius $n/4$ around the identity element. We prove that R. Thompson groups $F$, $T$, $V$ have linear divergence functions. Therefore the asymptotic cones of these groups do not have cut points.

Fall 2018

2019-02-18T08:49:06-04:00

Fall 2018

August 30
Speaker: Matt Zaremsky (University at Albany)
Title: The Bieri-Neumann-Strebel-Renz invariants of the Houghton groups

Abstract: The Houghton groups $H_n$ are a family of groups that are straightforward to define but have a variety of bizarre and interesting properties. In this talk I will discuss my recent computation of the Bieri-Neumann-Strebel-Renz invariants $\Sigma^m(H_n)$ of the $H_n$. The computation reveals some geometry reminiscent of that expected for metabelian groups by Bieri's $\Sigma^m$-Conjecture, and has implications for the finiteness properties of certain subgroups of $H_n$. This talk will be self-contained, and I will not assume any particular familiarity with the Houghton groups or the BNSR-invariants.

September 6
Speaker: Jonathan Williams (Binghamton)
Title: DGAs and Legendrian knots

Abstract: In order to discuss recent work, I will introduce Legendrian contact homology as formulated by Chekanov.

September 13
Speaker: Julie Bergner (University of Virginia)
Title: An introduction to 2-Segal sets via combinatorial examples

Abstract: The notion of a 2-Segal object was recently defined by Dyckerhoff and Kapranov, and independently by Gálvez-Carrillo, Kock, and Tonks under the name of decomposition space. Whereas 1-Segal sets model the structure of a category, in which composition is defined and is associative, 2-Segal sets instead encode a more general structure in which composition need not exist or be unique, but is still associative when it is defined. The 2-Segal set associated to a graph gives a nice example where maps can be composed in different ways. In particular, following a definition of Dyckerhoff and Kapranov, this 2-Segal set has an associated Hall algebra which is much smaller than most natural examples of such algebras and has a curious description as a cohomology ring.

September 27
Speaker: Jonathan Williams (Binghamton)
Title: Working topology seminar: DGAs in low-dimensional topology II

Abstract: I will continue my discussion of the Chekanov algebra and perhaps introduce my own recent work.

October 18
Speaker: Moshe Cohen (Vassar)
Title: Random knotting and billiard table diagrams

Abstract: We begin with a brief introduction to knot theory and proceed to a discussion of various models for randomness used to study knots. We then introduce a truncated model that allows us to get explicit probability formulae.

Koseleff and Pecker show that all knots can be parametrized by Chebyshev polynomials in three dimensions. These long knots can be realized as trajectories on billiard table diagrams. We use this knot diagram model to study random knot diagrams by flipping a coin at each 4-valent vertex of the trajectory.

We truncate this model to study 2-bridge knots together with the unknot. We give the exact probability of a knot arising in this model. Furthermore, we give the exact probability of obtaining a knot with crossing number c.

This is joint work with Chaim Even-Zohar and Sunder Ram Krishnan.

October 25
Speaker: Jake Blomquist (Binghamton)
Title: Bousfield-Kan Completion for Integral Chains, Iterated Suspension, and Stabilization

Abstract: Homology groups and homotopy groups of spaces are two of the main invariants used in algebraic topology to translate questions about spaces into the more rigid and computable setting of algebra. Homology groups are obtained by throwing away information, they are a form of derived abelianization of spaces, making them easier to compute than homotopy groups, but at the cost of losing information. Exploring connections between these invariants, such as the Hurewicz map, are thus very useful in helping to understand homotopy groups. One could ask the question: How much of a space does homology see? I will begin this talk by describing the classical completion of a space with respect to ordinary homology (with coefficients in a ring)—this construction goes back to Sullivan and Bousfield-Kan and provides an answer to this question.

One consequence is that in the case of integral completion, these constructions can be understood as arising from a comparison between spaces and simplicial abelian groups equipped with extra coalgebraic structure. In joint work with John E. Harper, we show that the integral chains functor fits into an equivalence of homotopy theories between 1-connected spaces and certain coalgebras of simplicial abelian groups; thus capturing the homotopy category of simply connected spaces as abelian complexes with additional coalgebraic structure.

The completion construction and derived equivalence proof ideas, exploiting Goodwillie's higher Blakers-Massey theorems, are sufficiently homotopical to establish similar results in several closely related comparisons including (i) suspension, (ii) iterated suspension, and (iii) stabilization of spaces.

Time permitting I'll also discuss how these ideas can be extended to the unstable context of structured ring spectra.

November 1
Speaker: Ross Geoghegan (Binghamton)
Title: The semistability problem for CAT(0) groups

Abstract: Proper CAT(0) spaces are natural generalizations of simply connected complete manifolds of non-positive sectional curvature. These spaces have a simple and beautiful metric geometry, and their boundaries at infinity are metric compacta. When a group $G$ acts properly discontinuously and cocompactly on a proper CAT(0) space (and this happens in a variety of important mathematical situations) $G$ is called a CAT(0) group. In this situation the point-set topology of the compact boundary reflects algebraic properties of the group. So this part of geometric group theory is a natural meeting place for algebra and topology. I will discuss the semistability problem for CAT(0) groups, a still-open problem which nicely illustrates this meeting place.

November 8
Speaker: Olakunle Abawonse (Binghamton)
Title: Discrete Morse Theory, Collapsibility and CAT(0) Simplicial Complexes

Abstract: We will talk about a sufficient condition under which a finite simplicial complex of dimension three or less and equipped with a piecewise Euclidean geometry collapses to a point according to a result of Katherine Crowley. This will be done using discrete Morse theory, a combinatorial analog to the classical smooth theory.

November 15
Speaker: C S Aravinda (TIFR Centre for Applicable Mathematics, Bangalore)
Title: Geodesic conjugacies in nonpositive curvature

Abstract: The question of whether a time-preserving geodesic conjugacy determines a closed, negatively curved Riemannian manifold up to an isometry is one of the central problems in Riemannian geometry. While an answer to the question in this generality has yet remained elusive, this talk will briefly give an overview and discuss a certain improvement of a known result.

Fall 2019

2020-01-27T08:20:16-04:00

Fall 2019

August 22
Speaker: TBA (Institution)
Title: TBA

Abstract: TBA

August 29
Speaker: Catherine Pfaff (Queen's University)
Title: Typical Trees: An $Out(F_r)$ Excursion

Abstract: Random walks are not new to geometric group theory (see, for example, work of Furstenberg, Kaimonovich, Masur). However, following independent proofs by Maher and Rivin that pseudo-Anosovs are generic within mapping class groups, and then new techniques developed by Maher-Tiozzo, Sisto, and others, the field has seen in the past decade a veritable explosion of results. In a 2 paper series, we answer with fine detail a question posed by Handel-Mosher asking about invariants of generic outer automorphisms of free groups and then a question posed by Bestvina as to properties of $\mathbb R$-trees of full measure in the boundary of Culler-Vogtmann outer space. This is joint work with Ilya Kapovich, Joseph Maher, and Samuel J. Taylor.

September 5
Speaker: TBA (Institution)
Title: TBA

Abstract: TBA

September 12
Speaker: Caglar Uyanik (Yale University)
Title: Dynamics on geodesic currents and atoroidal subgroups of $Out(F_N)$

Abstract: Geodesic currents on surfaces are measure theoretic generalizations of closed curves on surfaces and they play an important role in the study of the Teichmuller spaces. I will talk about their analogs in the setting of free groups, and try to illustrate how the dynamics and geometry of the $Out(F_N)$ action reflects on the algebraic structure of $Out(F_N)$.

September 19
Speaker: Cary Malkiewich (Binghamton University)
Title: What algebraic K-theory has to do with fixed-point theory

Abstract: The goal of this talk is to give a gentle introduction to algebraic $K$-theory and the Dennis trace. We'll see how the concept naturally arises when we try to enumerate all the ways to algebraically count the fixed points of a map $f: X \rightarrow X$ for a finite CW complex $X$.

September 26
Speaker: TBA (Institution)
Title: TBA

Abstract: TBA

October 3
Speaker: TBA (Institution)
Title: TBA

Abstract: TBA

October 10
Speaker: Inbar Klang (Columbia University)
Title: Hochschild homology for $C_n$-equivariant things

Abstract: After introducing Hochschild homology and topological Hochschild homology, I will talk about about the twisted versions of these that can be defined in the presence of an action of a finite cyclic group. I will discuss joint work with Adamyk, Gerhardt, Hess, and Kong in which we develop a theoretical framework and computational tools for these twisted Hochschild homology theories.

SPECIAL DATE AND TIME: October 15, 1:15 - 2:15, WH 100E (Joint with the Combinatorics Seminar)
Speaker: Emanuele Delucchi (Fribourg/Freiburg)
Title: Fundamental Polytopes of Metric Spaces via Parallel Connection of Matroids

Abstract: Motivated by applications in phylogenetics, Linard Hoessly and I tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010. We consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid. We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics, and we characterize the metric trees for which the fundamental polytope is simplicial.

October 17
Speaker: TBA (Institution)
Title: TBA

Abstract: TBA

October 24
Speaker: Nicholas Vlamis (CUNY, Queen's College and Graduate Center)
Title: Topology of (big) mapping class groups

Abstract: Mapping class groups inherit a natural topology from the compact-open topology on homeomorphism groups. When the underlying surface is of infinite type, this topology is no longer discrete, which allows us to study these mapping class groups from the perspective of topological group theory. We will explain how to see that mapping class groups are Polish groups and how we can use topological aspects to prove algebraic statements.

SPECIAL DATE AND TIME: October 28, 4:30 - 5:30, WH 100E
Speaker: Yash Lodha (EPFL)
Title: Property FW and smoothability

Abstract: I shall describe joint work with Matte Bon and Triestino. We demonstrate that aperiodic actions of Kazhdan groups by countably singular diffeomorphisms on closed manifolds are smoothable. In the case of the circle, we obtain a proof that groups of piecewise linear or piecewise projective homeomorphisms are not Kazhdan unless they are finite. The key new idea is the application of Property FW, which is a weakening of Kazhdan's property (T).

October 31
Speaker: Eduard Schesler (Universität Bielefeld)
Title: The Sigma conjecture for solvable $S$-arithmetic groups via discrete Morse theory on Euclidean buildings.

Abstract: Given a finitely generated group $G$, the $\Sigma$ invariants of $G$ consist of geometrically defined subsets $\Sigma^k(G)$ of the set $S(G)$ of all characters $\chi: G\to \mathbf{R}$ of $G$. These invariants were introduced independently by Bieri-Strebel and Neumann for $k=1$ and generalized by Bieri-Renz to the general case in the late 80's in order to determine the finiteness properties of all subgroups $H$ of $G$ that contain the commutator subgroup $[G,G]$. In this talk we determine the Sigma invariants of certain $S$-arithmetic subgroups of Borel groups in Chevalley groups. In particular we will determine the finiteness properties of every subgroup of the group of upper triangular matrices $B_n(\mathbf{Z}[1/p]) < SL_n(\mathbf{Z}[1/p])$ that contains the group $U_n(\mathbf{Z}[1/p])$ of unipotent matrices where $p$ is any sufficiently large prime number.

November 7
Speaker: Edgar Bering (Temple University)
Title: Special covers of alternating links

Abstract: The “virtual conjectures” in low-dimensional topology, stated by Thurston in 1982, postulated that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved in 2012 by Agol and Wise, using the machine of special cube complexes. Since that time, many mathematicians have asked how big a cover one needs to take to ensure one of these desired properties.

We begin to give a quantitative answer to this question, in the setting of alternating links in $S^3$. If an alternating link L has a diagram with n crossings, we prove that the complement of L has a special cover of degree less than $72((n-1)!)^2$. As a corollary, we bound the degree of the cover required to get Betti number at least $k$. We also quantify residual finiteness, bounding the degree of a cover where a closed curve of length $k$ fails to lift. This is joint work with David Futer.

November 14
Speaker: Ian Frankel (Queens University)
Title: Quantitative recurrence and hyperbolicity in Teichmüller space

Abstract: The moduli space of compact hyperbolic surfaces of genus $g > 1$ has a metric which displays geometric and dynamical properties similar to the properties of hyperbolic surfaces. In particular, we will describe a Teichmüller space analog of the basic fact from hyperbolic geometry: if two geodesics in hyperbolic space converge to the same point on the boundary, then the distance between them decreases exponentially.

November 21
Speaker: Yu Zhang (Ohio State University)
Title: Topological Quillen localization of structured ring spectra

Abstract: Homotopy groups and stable homotopy groups of spaces are central invariants in algebraic topology. Stable homotopy groups are comparatively easier to work with, at the expense of losing certain information. However, if we are working with nice spaces, nothing will be lost by working stably: A map between nilpotent spaces induces homotopy groups isomorphisms if and only if it induces stable homotopy groups isomorphisms.

Structured ring spectra are spectra with certain algebraic structure encoded by the action of an operad O. For such O-algebras, the analog of stable homotopy groups are played by Topological Quillen (TQ) homology groups. In this talk, we will draw the analogy between topological spaces and O-algebras, discuss the TQ-localization of O-algebras, and show the TQ-Whitehead theorem for homotopy pro-nilpotent O-algebras. Part of the work in this talk is joint with John E. Harper.

SPECIAL DATE AND TIME: November 26, 1:15 - 2:15, WH 100E (Joint with the Combinatorics Seminar)
Speaker: Casey Donoven (Binghamton University)
Title: Counting Boxes: A Friendly Introduction to Fractal Dimension

Abstract: The box-counting dimension of a set is calculated by covering a set with 'boxes' and seeing how fast the number of boxes grows in proportion to decreasing the size of the box. For manifolds in Euclidean space, such as curves, surfaces, etc., the box-counting dimension agrees with the topological dimension. My work explores the box-counting dimension (and Hausdorff dimension) in Cantor space, the boundary of an infinite tree. Specifically, I will answer the following question: Given sets $E$ and $F$ in Cantor space and a random isometry $f$, what is the dimension of the intersection of $E$ and $f(F)$?

November 28

No seminar (Thanksgiving)

December 5
Speaker: Steven Gindi (Binghamton University)
Title: Long Time Limits of Generalized Ricci Flow

Abstract: We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles. This leads to rigidity and classification results for nonsingular solutions.

Fall 2020

2021-08-26T16:27:23-04:00

Fall 2020

September 10
Speaker: Abdalrazzaq Zalloum (Queen's University)
Title: Hyperbolic-like boundaries of non-hyperbolic spaces.

Abstract: While a quasi-isometry between two hyperbolic spaces X,Y induces a homeomorphism between their respective Gromov boundaries, the conclusion fails if X,Y are replaced by cocompact CAT(0) spaces. This has motivated a bulk of recent work introducing “hyperbolic-like” boundaries for CAT(0) spaces, and more generally, for proper geodesic metric spaces. For a proper geodesic metric space X, instead of considering the collection of all geodesic rays shooting to infinity, if you collect only those possessing some “hyperbolic-like” behavior, you obtain a boundary which is invariant under-quasi isometries. I will describe few such boundaries along with the way they relate to each other. Some of the results I will mention are joint with Incerti-Medici while others are joint with Qing and Murray.

September 17
Speaker: Mark Pengitore (OSU)
Title: Coarse embeddings and homological filling functions

Abstract: In this talk, we will relate homological filling functions and the existence of coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result.

September 24
Speaker: Matt Haulmark (Binghamton University)
Title: Introduction to relatively hyperbolic groups and Bowditch boundaries

Abstract: The notion of a group being hyperbolic relative to class of subgroups was introduced by Gromov to generalize word hyperbolic and geometrically finite Kleinian groups. Associated to a group with a relatively hyperbolic structure is a compact metric space called the Bowditch boundary. The topology of this boundary can provide information about the group. In this talk, we will introduce relatively hyperbolicity for finitely generated groups. We will also discuss the Bowditch boundary and survey some results. The goal of this lecture is to prepare the audience for Ashani Dasgupta's talk the following week.

October 1
Speaker: Ashani Dasgupta (UW-Milwaukee)
Title: Local connectedness of Bowditch Boundary

Abstract: Bowditch associated a topological space ∂G to Relatively Hyperbolic Group G. Topological information about ∂G is often useful to understand algebraic information about the group G. In this talk we will sketch a proof of the following theorem: if G is a finitely generated, relatively one-ended, relatively hyperbolic group then ∂G is locally connected. Earlier Bowditch proved the local connectedness of ∂G with a more restricted hypothesis. We will sketch the proof of the general result.

October 8
Speaker: Kathryn Mann (Cornell University)
Title: Reconstructing maps out of groups

Abstract: This talk is about the relationship between the algebraic structure of a discrete group and the possible dynamics of its actions. I'll explain a result with Maxime Wolff where we show that, under some circumstances, one can completely reconstruct a homeomorphism of a space out of algebraic (group structure) data. As a consequence of this, we gave an independent short proof of a recent theorem of Kim and Koberda: you can distinguish the group of $C^r$ diffeomorphisms of a 1-manifold from the group of $C^s$ diffeomorphisms just by knowing their finitely generated subgroups.

October 15
Speaker: Daniel Studenmund (Binghamton University)
Title: Vanishing in top-dimensional cohomology of $GL_n(\mathcal{O})$

Abstract: In this talk, we will address one instance of the general question: Given a group $G$, what is the cohomology of $G$ with rational coefficients? A celebrated result of Borel and Serre implies that $G = GL_n(\mathcal{O})$ has finite virtual cohomological dimension (vcd) when $\mathcal{O}$ is the ring of integers of a number field $K$. This implies that cohomology with rational coefficients vanishes in dimensions greater than the vcd, but leaves open the question of whether cohomology vanishes in the vcd. After surveying some results in the area, I will discuss joint work with Andy Putman which computes cohomology of $GL_n(\mathcal{O})$ in the vcd for certain $\mathcal{O}$, and how there is a surprisingly subtle dependence on the number ring $\mathcal{O}$.

October 22
Speaker: Francisco Arana Herrera (Stanford University)
Title: Counting hyperbolic multi-geodesics with respect to the lengths of individual components.

Abstract: In her thesis, Mirzakhani showed that on any closed hyperbolic surface of genus $g$, the number of simple closed geodesics of length at most $L$ is asymptotic to a polynomial in $L$ of degree $6g-6$. Wolpert conjectured that analogous results should hold for more general countings of multi-geodesics that keep track of the lengths of individual components. In this talk we will present a proof of this conjecture which combines techniques and results of Mirzakhani with ideas introduced by Margulis in his thesis.

October 29
Speaker: Rylee Lyman (Rutgers)
Title: Nielsen realization for infinite-type surfaces

Abstract: We learn the classification of surfaces early in our mathematical careers: the homeomorphism type of an orientable surface with finitely generated fundamental group is determined by genus, punctures and boundary components. Without the finite generation assumption, there is still a classification, due to Kerékjártó and Richards. These surfaces are of infinite type. Associated to any surface is its mapping class group. A famous theorem of Kerckhoff from 1983 solves the “Nielsen realization” problem posed in 1932: finite subgroups of the mapping class group of a finite-type surface of negative Euler characteristic are exactly the groups of isometries of some hyperbolic metric on the surface. Recently, joint with Santana Afton, Danny Calegari and Lvzhou Chen, I extended Kerckhoff's theorem to orientable, infinite-type surfaces. I'd like to introduce infinite-type surfaces and discuss the theorem and some of its consequences.

November 5
Speaker: Ignat Soroko (LSU)
Title: Groups of type FP: their quasi-isometry classes and homological Dehn functions

Abstract: There are only countably many isomorphism classes of finitely presented groups, i.e. groups of type $F_2$. Considering a homological analog of finite presentability, we get the class of groups $FP_2$. Ian Leary proved that there are uncountably many isomorphism classes of groups of type $FP_2$ (and even of finer class FP). R.Kropholler, Leary and I proved that there are uncountably many classes of groups of type FP even up to quasi-isometries. Since `almost all' of these groups are infinitely presented, the usual Dehn function makes no sense for them, but the homological Dehn function is well-defined. In an on-going project with N.Brady, R.Kropholler and myself, we show that for any integer $k\ge4$ there exist uncountably many quasi-isometry classes of groups of type FP with a homological Dehn function $n^k$. In this talk I will give the relevant definitions and describe the construction of these groups. Time permitting, I will describe the connection of these groups to the Relation Gap Problem.

November 19
Speaker: Bena Tshishiku (Brown University)
Title: Arithmeticity of free-abelian by cyclic groups

Abstract: We discuss arithmeticity of groups G(A) = ℤ^n ⋊ ℤ, where ℤ acts on ℤ^n by powers of an irreducible, hyperbolic matrix A ∈ GL(n,ℤ). The question of when G(A) is arithmetic was studied systematically by Grunewald-Platonov, but there are basic things that we still don't know. For example, for what values of n is there an arithmetic example? We discuss some progress toward answering this question.

December 3
Speaker: Hakan Doga (SUNY Buffalo)
Title: A Combinatorial Description of the Knot Concordance Invariant Epsilon

Abstract: The knot concordance group is one of the central objects in the study of topological knot types and low-dimensional topology. Concordance invariants obtained from knot Floer homology have resulted in some important classification results. In this talk, I will describe a new method to compute the concordance invariant epsilon using combinatorial knot Floer homology and show the computations for torus knots and positive braids. This is a joint work with S. Dey.

Fall 2021

2021-12-10T11:37:35-04:00

Fall 2021

September 2nd
Speaker: Daniel Studenmund (Binghamton)
Title: Homotopy equivalences of full solenoids

Abstract: Solenoids, inverse limits of self-coverings of the circle, are important examples of compact connected metrizable spaces. They were studied by topologists Mayer and van Dantzig, and arise in the theory of hyperbolic dynamical systems as the Smale-Williams attractor. We will use ideas from shape theory to show that homotopy equivalences of a solenoid naturally correspond to certain rational numbers. The full solenoid over a space X is the inverse limit of -all- finite covers of X. We will state a generalization of the 1-dimensional result, relating homotopy equivalences of the full solenoid over a finite CW complex X to isomorphisms between finite-index subgroups of pi_1(X). If time permits, we will say a word on the Teichmuller theory of the full solenoid over a closed hyperbolic surface.

September 9th
Speaker: José Román Aranda Cuevas (Binghamton)
Title: Trisecting objects in dimension four

Abstract: In 2012, D. Gay and R. Kirby proved that every closed oriented 4-manifold admits a trisection: a decomposition of the space into three standard pieces. Since then, many mathematicians have used trisections to study 4-manifolds from various perspectives: from morse functions, complexes of curves, group theory, to mention some. The goal of this talk is to survey recent ideas and results surrounding the theory of trisections. No specialized background will be assumed.

September 23rd
Speaker: Michael Dobbins (Binghamton)
Title: A strong equivariant deformation retraction from the homeomorphism group of the projective plane to the special orthogonal group

Abstract: I will present the construction of a strong G-equivariant deformation retraction from the homeomorphism group of the 2-sphere to the orthogonal group, where G acts on the left by isometry and on the right by reflection through the origin. This induces a strong G-equivariant deformation retraction from the homeomorphism group of the projective plane to the special orthogonal group, where G is the special orthogonal group acting on the projective plane. The same holds for subgroups of homeomorphisms that preserve the system of null sets. This confirms a conjecture of Mary-Elizabeth Hamstrom.

September 30th
Speaker: Ulysses Alvarez (Binghamton)
Title: The topology of a corank 1 matroid over $\Phi$

Abstract: Topological posets allow for the construction of a space which can be viewed as a generalization of the order complex of a discrete poset. We will discuss how this structure can be used to understand the topology of a corank 1 matroid over the tropical phase hyperfield on 4 elements.

October 7th
Speaker: Cary Malkiewich (Binghamton)
Title: The higher characteristic polynomial

Abstract: In this talk I will discuss various lifts of the characteristic polynomial to the setting of algebraic K-theory, and describe the relationship to trace methods and to topological fixed-point theory.

October 21st
Speaker: Nima Rasekh (EPFL) (virtual talk, streamed in WH100E)
Title: THH and Shadows of Bicategories

Abstract: Topological Hochschild homology (THH), first defined for ring spectra and then later dg-categories and spectrally enriched categories, is an important invariant with connections to algebraic K-theory and fixed point methods. The existence of THH in such diverse contexts motivated Ponto to introduce a notion that can encompass the various perspectives: a shadow of bicategories. On the other side, many versions of THH have been generalized to the homotopy coherent setting providing us with motivation to develop an analogous homotopy coherent notion of shadows.

The goal of this talk is to use an appropriate bicategorical notion of THH to prove that a shadow on a bicategory is equivalent to a functor out of THH of that bicategory. We then use this result to give an alternative conceptual understanding of shadows as well as an appropriate definition of a homotopy coherent shadow.

This is joint work with Kathryn Hess.

October 28th
Speaker: Jonathan Williams (Binghamton)
Title: Turning a Lefschetz fibration into a crown map

Abstract: A rich source of examples of smooth 4-manifolds comes from finding a composition of Dehn twists on a closed surface which is isotopic to the identity map. I'll describe how to turn this into a source of examples of crown diagrams of smooth 4-manifolds.

November 4th
No seminar this week

November 11th
Speaker: Tim Susse (Bard) (virtual talk, streamed in WH100E)
Title: When is a RACG QI to a RAAG: a probabilistic approach.

Abstract: A celebrated theorem of Davis and Januszkiewicz shows that every right-angled Artin group (RAAG) is isomorphic to a finite index subgroup of some right-angled Coxeter group (RACG). The converse, however, is not true and the question of which RACGs are quasi-isometric to RAAGs has achieved folk status. In this talk we will discuss the state of the art on this question, which uses some of the most powerful tools in Geometric Group Theory. We will focus on the generic version of this question, using random graphs to model random right-angled Coxeter groups and show that at low enough density the answer is (almost surely) never.

November 18th
No seminar this week (cancellation)

December 2nd
Speaker: Jonathan Williams (Binghamton)
Title: The salient sequence of a crown diagram

Abstract: I will discuss a very concrete and elementary construction allowing one to associate a pair of numbers to each crossing in a crown diagram, and discuss invariance properties for a particular “salient sequence” of these numbers. If time permits, I'll point out a few promising directions in which one could take the construction.

Fall 2022

2023-01-17T11:57:51-04:00

Fall 2022

August 25th
Speaker: Cary Malkiewich (Binghamton)
Title: A Farrell-Jones isomorphism for scissors congruence K-theory

Abstract: Scissors congruence K-theory is an algebraic object that captures solutions to variants of Hilbert's Third Problem, in other words, when one polytope can be cut into pieces and rearranged to form another. In this talk I will describe a new trace map from scissors congruence K-theory to group homology. It turns out that a refinement of this map provides an inverse to the assembly map, proving the Farrell-Jones isomorphism for this form of K-theory. This allows us to make the first computations of scissors congruence K-theory groups above K_1. Much of this is joint work with Mona Merling and Inna Zakharevich.

September 1st
Speaker: Matt Haulmark (Binghamton)
Title: Cube Complexes and Stuff

Abstract: Much of this talk will be introductory. Some of the topics covered will be CAT(0) cube complexes, wallspaces, and the CAT(0) cube complex dual to a wallspace. Then I will briefly discuss work in progress with Jason Manning.

September 8th
Speaker: William Menasco (University at Buffalo)
Title: Surface Embeddings in $\mathbb{R}^2 \times \mathbb{R}$

Abstract: In this joint work with Margaret Nichols, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $\pi : \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}^2$ be the natural projection map onto the Euclidean plane. Let $ \epsilon : S_g \rightarrow \mathbb{R}^2 \times \mathbb{R}$ be a smooth embedding of a closed oriented genus $g$ surface such that the set of critical points for the map $\pi \circ \epsilon$ is a piece-wise smooth (possibly multi-component) $1$-manifold, $\mathcal{C} \subset S_g$. We say $\mathcal{C}$ is the {\em crease set of $\epsilon$} and two embeddings are in the same {\em isotopy class} if there exists an isotopy between them that has $\mathcal{C}$ being an invariant set. The case where $\pi \circ \epsilon |_\mathcal{C}$ restricts to an immersion is readily accessible, since the turning number function of a smooth curve in $\mathbb{R}^2$ supplies us with a natural map of components of $\mathcal{C}$ into $\mathbb{Z}$. The Gauss-Bonnet Theorem beautifully governs the behavior of $\pi \circ \epsilon (\mathcal{C})$, as it implies $\chi(S_g) = 2 \sum_{\gamma \in \mathcal{C}} t( \pi \circ \epsilon (\gamma))$, where $t$ is the turning number function. Focusing on when $S_g \cong S^2$, we give a necessary and sufficient condition for when a disjoint collection of curves $\mathcal{C} \subset S^2$ can be realized as the crease set of an embedding $\epsilon: S^2 \rightarrow \mathbb{R}^2 \times \mathbb{R}$.

September 15th
Speaker: Gage Martin (MIT) Zoom talk
Title: Annular links, double branched covers, and annular Khovanov homology

Abstract: Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.

September 22nd
Speaker: Alexandra Kjuchukova (Notre Dame) Zoom talk
Title: Coxeter groups and bridge numbers of links

Abstract: The 1970s meridional rank conjecture of Cappell and Shaneson posits equality between two invariants of links in S^3, the bridge number and meridional rank. I will define these invariants and outline the history of the conjecture. Then, I will describe an approach which relies on finding Coxeter quotients of the fundamental groups of link complements. I will use this approach to establish the conjecture for certain infinite families of links, as well as to derive explicit formulas for the bridge numbers for the links in question. Based on joint works with Blair, Baader, Misev.

September 29th
Speaker: J Williams (Binghamton)
Title: Nullhomotopic crown maps

Abstract: I'll present a straightforward way to convert any crown map (which is a special kind of map from a 4-manifold to the 2-sphere) into a nullhomotopic crown map, while keeping track of the combinatorial data of the original map. The construction will also bring salient sets into the very active world of trisections and multisections.

October 6th
Speaker: Gary Guth (University of Oregon) Zoom talk
Title: Satellites, Stabilizations, and Exotic Surfaces

Abstract: A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable”. For example, in thinking about the four-dimensional h-coboridism theorem, Wall proved that simply connected, exotic four-manifolds always become smoothly equivalent after applying a suitable stabilization operation enough times. Similarly, Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. The question remains, “how many stabilizations are necessary, and is one always enough?” By considering certain satellite operations, we provide an answer to this question in the case of exotic surfaces with boundary.

October 13th
Speaker: Rhiannon Griffiths (Cornell)
Title: Slices of higher categories

Abstract: Fully weak higher categories are often the most useful for applications to areas such as algebraic topology and homotopy theory, but become too complex for practical use in dimensions greater than 2. A solution is to find a notion of semi-strict higher category: higher categories which are just weak enough to be equivalent to the fully weak variety, while still being tractable enough to work with directly.

In this talk I will show how it is possible to take ‘slices’ of higher categories. Roughly speaking, the k-th slice is the symmetric operad corresponding to the algebra formed by the k-cells of a higher category. I will explain the ways in which these slices can give us a way to recognise when some notion of higher category is equivalent to the fully weak variety, and discuss some examples and potential applications.

October 20th
Fall Break

October 27th
Speaker: Prayagdeep Parija (UW-Milwaukee) Zoom talk
Title: Random quotients of hyperbolic groups and Property (T)

Abstract: What does a random quotient of a group look like? Gromov introduced the density model of quotients of free groups. The density parameter d measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he proved that for d<1/2 a random quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov's result to show that for d<1/2 a random quotient of even a non-elementary hyperbolic group is non-elementary hyperbolic.

Żuk/Kotowski-Kotowski proved that for d>1/3 a typical quotient of a free group has Property (T). We show that for 1/3<d<1/2 (in a closely related density model) a random quotient of a non-elementary hyperbolic group is non-elementary hyperbolic and has Property-(T).

This provides an answer to a question of Gromov (and Ollivier)

November 3rd
Speaker: Thomas Brazelton (Penn)
Title: Equivariant enumerative geometry

Abstract: Classical enumerative geometry asks geometric questions of the form “how many?” and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of “equivariant enumerative geometry”, which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the sum of regular representations of the orbits of solutions to an equivariant enumerative problem are conserved.

November 10th
Speaker: Meenakshy Jyothis (Binghamton)
Title: Automorphisms of geodesic currents the preserves intersection form

Abstract: In this talk we will be looking at Ivanov's conjecture in the context of geodesic currents. The space of geodesic currents generalizes various objects of interest on a surface, such as the set of closed curves up to homotopy and the Teichmüller space. I will talk about a particular group of automorphisms of this space and will compare it to the mapping class group.

Invanov's conjectured that every object naturally associated to a surface and having a `sufficiently rich' structure has mapping class group as its groups of automorphisms. It is already known that the conjecture holds true for various combinatorial objects associated with a surface as well as for the Teichmüller space of a surface.

November 17th
No seminar this week

November 24th
Thanksgiving Break

December 1st
Speaker: Nobukata Asano (National Institute of Technology, Tsuyama College) Zoom talk
Title: Some lower bounds for the Kirby-Thompson invariant of 4-manifolds

Abstract: A trisection is a decomposition of a closed 4-manifold X into a 3-tuple of 4-dimensional 1-handlebodies, which was introduced by Gay and Kirby. Kirby and Thompson defined an invariant of X by using trisections. This invariant is called the Kirby-Thompson invariant. In this talk, we give some lower bounds for the Kirby-Thompson invariant of certain 4-manifolds. As an application, we find the first example of a 4-manifold whose Kirby-Thompson invariant is non-zero. This is a joint work with Hironobu Naoe (Chuo University) and Masaki Ogawa (Saitama University).

December 8th
Speaker: Lea Kenigsberg (Columbia)
Title: Coproduct structures, a tale of two outputs

Abstract: I will tell the elusive story of coproduct structures in Floer theory and string topology, and explain why we care about them. I will then define a new coproduct structure on the symplectic cohomology of Liouville manifolds. Time permitting, I will indicate how to compute it in an example to show that this structure is not trivial. This will be an accessible talk; I will not assume knowledge of Floer theory or string topology. This is based on my thesis work, in progress.

Fall 2023

2024-01-10T09:28:17-04:00

Fall 2023

August 31st
No seminar this week

September 7th
No seminar this week (cancellation)

September 14th
Speaker: John Rached (Binghamton)
Title: Quantitative behavior of horocycle flow on the moduli space of genus 2 surfaces

Abstract: The action of SL(2,R) on moduli space exhibits measure rigidity, analogously to Ratner’s theorems for unipotent flows on homogeneous spaces, due to the seminal work of Eskin-Mirzakhani. Similar results cannot hold for the horocycle flow on moduli space, but for special subvarieties of strata (eigenform loci), some key tools from homogeneous dynamics have an incarnation in this inhomogeneous setting. A version of Ratner’s theorem holds for eigenform loci, and a flurry of recent work on quantitative results for actions on homogeneous spaces begs a natural question - can one effectivize arguments for the horocycle flow on eigenform loci? We give some support for a positive answer to this question, and make some conjectures.

September 21st
No seminar this week (cancellation)

September 28th
Speaker: Maxine Calle (University of Pennsylvania)
Title: Nested cobordisms and TQFTs

Abstract: The folk theorem identifying 2-dimensional TFQTs with Frobenius algebras is a starting point for a lot of interesting mathematics, from mathematical physics to homotopy theory to higher category theory. In this talk, we will explore what happens if we replace the cobordism category with a category of nested cobordisms, where 2-dimensional surfaces may have embedded 1-dimensional submanifolds, and what kind of algebraic structure the corresponding nested TQFTs pick out. This is based on ongoing work joint with R. Hoekzema, L. Murray, N. Pacheco-Tallaj, C. Rovi, and S. Sridhar.

October 5th
Speaker: Cary Malkiewich (Binghamton)
Title: Higher scissors congruence

Abstract: Hilbert's Third Problem asks for sufficient conditions that determine when two polyhedra in three-dimensional Euclidean space are scissors congruent. Classically, the attempts to solve this problem (in this and other geometries) lead into group homology and algebraic K-theory, in a somewhat ad-hoc way. In the last decade, Zakharevich has shown that the presence of K-theory here is not ad-hoc, but is integral to the definition of scissors congruence itself. This leads to a natural notion of higher scissors congruence groups, in which the 0th group is the classical one that determines the answer to Hilbert's Third Problem.

In this talk, I'll describe a surprising recent result that these higher groups arise from a Thom spectrum. Its base space is the homotopy orbit space of a Tits complex, and the vector bundle is the negative tangent bundle of the underlying geometry. Using this result, we can explicitly compute the higher scissors congruence groups for the one-dimensional geometries, and give exact sequences that express them for the two-dimensional geometries. Much of this is joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich.

October 12th
Speaker: Jenya Sapir (Binghamton)
Title: Geometry of geodesic currents

Abstract: The space of projective, filling currents PFC(S) contains many structures relating to a closed, genus g surface S. For example, it contains the set of all closed curves on S, as well as an embedded copy of Teichmuller space, and many other spaces of metrics on S. We will discuss a structure theorem that compares each filling current with a suitably chosen point in Teichmuller space. We will then use this structure theorem to explore the geometry of PFC(S) under an extension of the Thurston metric.

October 19th
No seminar this week (Fall break)

October 26th
Speaker: Brenda Johnson (Union College)
Title: What is (functor) calculus?

Abstract: Goodwillie’s calculus of homotopy functors is an important topological tool that has been used to shed light on and make connections between fundamental structures in homotopy theory and $K$-theory. It has also inspired the creation of new types of functor calculi to tackle problems in algebra and topology. In this talk, I will begin by describing Goodwillie’s calculus and some of these other types of functor calculi. I will then address more general questions about what the essential features of something called “functor calculus” should be and the types of conditions and ingredients that are sufficient for creating new functor calculi.

November 2nd
Speaker: Paul Apisa (University of Wisconsin)
Title: Hurwitz Spaces, Hecke Actions, and Orbit Closures in Moduli Space

Abstract: The moduli space of Riemann surfaces is a space whose points correspond to the ways to endow a surface with a hyperbolic metric or, equivalently, complex structure. Geodesic flow on moduli space can be used to generate an action of GL(2, R) on its cotangent bundle. While work of Eskin, Mirzakhani, Mohammadi, and Filip implies that GL(2, R) orbit closures are varieties, the question of which ones occur is wide open. Aside from two well-understood constructions (taking loci of branched covers and subloci of rank two orbit closures) there are only 3 known families of orbit closures: the Bouw-Moller curves, the Eskin-McMullen-Mukamel-Wright (EMMW) examples, and 2 sporadic examples. Building on ideas of Delecroix-Rueth-Wright, I will describe work showing that the Bouw-Moller and EMMW examples can be constructed using just the representation theory of finite groups. The main idea is to connect these examples to Hurwitz spaces of G-regular covers of the sphere for an appropriate finite group G. In the end, I will describe a construction that inputs a finite group G and a set of generators satisfying a combinatorial condition and outputs a GL(2, R) orbit closure in moduli space.

November 9th
No seminar this week

November 16th
Speaker: Andres Mejia (University of Pennsylvania)
Title: A Genuine Linearization Map for Equivariant Algebraic K-theory

Abstract: The Algebraic K-theory of a smooth manifold is a receptacle for many sensitive invariants. The driving example is the classical fact that the H-cobordism type of a manifold is completely controlled by only the fundamental group of its Algebraic K-Theory space. In fact, there is a reduction to a related invariant that only depends on the fundamental group of the manifold M. Turning to higher invariants, we are not so lucky, but there is still a comparison map called the linearization map that lets us compute parts of the Algebraic K-theory space in good situations. This talk will discuss a new construction of the linearization map when we are presented with a manifold together with the action of a finite group. If time permits, we will also discuss future directions with a view towards equivariant stable cobordism. These results are joint with D. Chan and M. Calle.

November 23rd
No seminar this week (Thanksgiving)

November 30th
No seminar this week

December 7th
Speaker: Collin Bleak (University of St Andrews) (virtual talk)
Title: On the maximal subgroups of R. Thompson's group V.

Abstract: The maximal subgroups of various groups have been a focus of study since the highly influential O'Nan–Scott Theorem of 1979, which classified the maximal subgroups of the finite symmetric groups. Motivated by our perspective on R. Thompson's group V as a natural generalisation of the finite symmetric and alternating groups to an infinite context, we have been exploring the maximal subgroup structure of V, working to move beyond the previously known maximal subgroups: the automorphic images of T and the set-wise stabilisers of finite sets of points in Cantor space (all with the same tail class).

We introduce the concept of a type system P, that is, a partition on the set of finite words over the alphabet {0,1} compatible with the partial action of Thompson's group V, and associate a subgroup Stab_V(P) of V. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of V. A byproduct of our approach is that we can specify an uncountable family of pairwise non-isomorphic maximal subgroups of V.

Joint with Jim Belk, Martyn Quick, and Rachel Skipper.

Fall 2024

2024-12-31T13:55:11-04:00

Fall 2024

August 22nd
Organizational meeting, meet in WH 100E at 2:50pm

August 29th
No seminar this week

September 5th
Speaker: Ezekiel Lemann (Binghamton University)
Title: Scissors Automorphism Groups and Homological Stability

Abstract: In this talk we will outline the proof of homological stability for scissors automorphism groups and highlight a number of consequences and related results. These include plus and group completion constructions for assembler K-theory, interpreting higher K-theory in terms of automorphism groups, and homology calculations for some scissors automorphism groups. The content of the talk is joint work with Kupers, Malkiewich, Miller, and Sroka.

September 12th
Speaker: Kimball Strong (Cornell University)
Title: Strictification of $\infty$-groupoids

Abstract: An $\infty$-groupoid is a mathematical object which generalizes a groupoid by possessing not just objects and morphisms, but also $2$-morphisms between morphisms, $3$-morphisms between $2$-morphisms, etc. Grothendieck conjectured that the category of $\infty$-groupoids up to homotopy was equivalent to the category of topological spaces up to homotopy, a still—unproven statement known as the “Homotopy Hypothesis.'' In this talk I will introduce a simpler object called a \textit{strict} $\infty$-groupoid, which encodes less information than an ordinary $\infty$-groupoid, but is easier to work with—much in the same way that homology groups are easier to work with than homotopy groups. I will then define a strictification functor that takes an ordinary $\infty$-groupoid and returns a strict $\infty$-groupoid, and prove that, up to homotopy, this functor can be used to encode the data of a topological space coalgebraically as a strict $\infty$-groupoid. Time permitting, I will discuss progress and open questions on generalizing this to $\infty$-categories.

September 19th
Speaker: Chase Vogeli (Cornell University)
Title: The Galois-equivariant $K$-theory of finite fields

Abstract: Abstract: Algebraic $K$-theory is an important invariant of rings defined using tools from homotopy theory. Recent progress in equivariant homotopy theory has enabled the study of equivariant algebraic $K$-theory for rings with actions by finite groups. In this talk, I will focus on the case of finite fields, where there is an action by their cyclic Galois groups. The nonequivariant $K$-groups of finite fields were computed by classical work of Quillen, and I will describe joint work with David Chan which extends this to a computation of the Galois-equivariant $K$-groups. In particular, we show the computation reduces to the well-studied coefficient groups of ordinary equivariant cohomology.

September 26th
Speaker: Lorenzo Ruffoni (Binghamton University)
Title: Atoroidal surface bundles with zero signature

Abstract: Hyperbolic geometry has been a powerful tool in the study of manifold topology. Beyond the classical theory of surfaces, Thurston showed that the family of surface bundles over the circle is a rich source of hyperbolic 3-manifolds. In dimension 4, the correct analogue is given surface bundles over surfaces. In order for such a bundle to admit a hyperbolic metric, it needs to satisfy some conditions, such as being atoroidal and having zero signature. Surprisingly enough, the first examples of atoroidal surface bundles over surfaces were constructed only a few months ago by Kent-Leininger. In this talk I’ll explain why these examples also have zero signature, meaning that they could admit hyperbolic metrics. This is joint work in progress with J-F. Lafont and N. Miller.

October 3rd
No seminar this week (Rosh Hashanah/Fall break)

October 10th
Speaker: Hanh Vo (Arizona State University)
Title: Curves with self-intersections on hyperbolic surfaces

Abstract: In this talk, I will discuss curves with self-intersections on hyperbolic surfaces. In the first part, I will speak about the k-systoles (where k is a natural number) of hyperbolic surfaces, which are the shortest closed geodesics with at least k self-intersections. In the second part, I will discuss how results regarding self-intersections of closed curves can be extended to arcs. This talk is based on joint work with A. Basmajian, M. Doan and H. Parlier.

October 17th
Speaker: Yu-Chan Chang (Wesleyan University)
Title: Graphical Properties of Bestvina–Brady Groups

Abstract: Given a finite simplicial graph, the associated right-angled Artin group has many properties that can be seen from the defining graph. For example, two right-angled Artin groups are isomorphic if and only if their defining graphs are isomorphic. Bestvina–Brady groups are exotic subgroups of right-angled Artin groups, and some algebraic structures of Bestvina–Brady groups can also be seen from the defining graphs. For instance, a Bestvina–Brady group is finitely generated if and only if the defining graph is connected. In this talk, I will discuss some properties of Bestvina–Brady groups from a graphical point of view. In particular, we will explore the Dehn functions and the RAAG recognition problem for Bestvina–Brady groups. Part of this talk is based on joint work with Lorenzo Ruffoni.

October 24th
Speaker: Kate Ponto (University of Kentucky)
Title: Iterated traces

Abstract: “Trace” usually evokes the trace of a matrix - so a number and so the idea of iterating feels a little strange. I'll discuss generalizations of the trace where taking the trace of a trace not only can be done, but feels almost inevitable.

October 31st
Speaker: Danika Van Niel (Binghamton University)
Title: Ghost Busting: How ghost maps helped solve a hard problem

Abstract: This spooky talk will discuss how to extend the classical concept of the affine line from algebraic geometry to equivariant algebra. The equivariant analogue to commutative rings are Tambara functors, classically the affine line is the spectrum of Z[x]. Equivariantly, this problem becomes significantly harder for two main reasons. First, G-Tambara functors are complex objects which send finite G-sets to commutative rings, and have maps between these rings. Second, when we adjoin a variable we can adjoin it to any level of the Tambara functor, even for the simple case of G = Cp we can add the variable at Cp/Cp or Cp/e. In this talk we will define Tambara functors, discuss these extended definitions, and demonstrate how the ghost map will make an extremely difficult question just difficult. This is joint work with David Chan, David Mehrle, J.D. Quigley, and Ben Spitz.

November 7th
Speaker: Shuchen Mu (Binghamton University)
Title: Transfer on Cyclic homology and Algebraic K-theory

Abstract: We will introduce a special kind of map called 'transfer' on Hochschild homology, cyclic homology and algebraic K-theory. And we will perform some calculation of examples.

November 14th
Speaker: Ismael Sierra (University of Toronto)
Title: Homological stability of even (and odd) symplectic groups

Abstract: I will define the “odd” symplectic groups $Sp_{2g+1}(\mathbb Z)$, which fit in between the usual “even” symplectic groups, and state new homological stability results for them. I will explain the main ideas of the proof and the sense in which the above can be seen as an algebraic analogue of the proof of homological stability of mapping class groups of surfaces by Harr–Vistrup–Wahl. Finally I will mention some related open questions.

November 21st
No seminar this week

November 28th
No seminar this week (Thanksgiving break)

December 5th
Speaker: Hari Rau-Murthy (University of Rochester)
Title: The Hopkins Kuhn Ravenel character and iterated THH

Abstract: This expository talk will talk about how to access the character of a representation of a finite group using Hochschild homology - a certain generalization of differential forms. We will present ideas for covering the Hopkins Kuhn Ravenel character- a generalization of the character of a representation - using this technology as well. Specifically, we will use a modified version of iterated Topological Hochschild Homology (THH). This work in progress is joint with Sanjana Agarwal, Noah Wisdom and Preston Cranford.

Fall 2025

2025-12-08T12:01:57-04:00

Fall 2025

August 28th
Speaker: Cary Malkiewich (Binghamton)
Title: Higher scissors congruence

Abstract: Scissors congruence is the study of polytopes, up to the relation of cutting into finitely many pieces and rearranging the pieces. In the 2010s, Zakharevich defined a “higher” version of scissors congruence, where we don't just ask whether two polytopes are scissors congruent, but also how many scissors congruences there are from one polytope to another.

Zakharevich's definition is a form of algebraic K-theory, which is famously difficult to compute, but I describe some recent work that makes these calculations possible, at least for low-dimensional geometries. This allows us to compute the homology of the group of cut-and-paste operations in new cases, including the group of interval exchange transformations, and a new proof of Szymik and Wahl's theorem that Thompson's group V is acyclic.

Much of this talk is based on joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich, and also with Alexander Kupers, Ezekiel Lemann, Jeremy Miller, and Robin Sroka.

September 4th
Speaker: Liz Tatum (Rochester)
Title: Some applications of equivariant Brown-Gitler spectra

Abstract: In the 1980s, Mahowald and Kane used integral Brown-Gitler spectra to construct splittings of the cooperations algebras for ko, connective real k-theory, and ku, connective complex k-theory. These splittings helped make it feasible to do computations using the ko- and ku-based Adams spectral sequences.

These spectral sequences have proven to be powerful tools for better understanding the structure of the stable homotopy groups of the sphere, with a variety of interesting applications. For example, Mahowald used them to verify the Telescope Conjecture at height one, and Gonzalez later used them to classify stunted lens spaces. In this talk, I will discuss progress towards developing analogues of these tools in the C_2 equivariant setting. In particular, Guchuan Li, Sarah Petersen, and I have constructed models for C_2-equivariant analogues of the integral Brown-Gitler spectra, and used them to construct an analogue of the ku-splitting.

September 18th
Speaker: Lorenzo Ruffoni (Binghamton)
Title: A Pontryagin sphere at infinity in real hyperbolic space

Abstract: Given a discrete group of isometries of hyperbolic space, we can look at its limit set, i.e., the set of accumulation points of its orbits on the sphere at infinity. This is a compact metric space embedded in the sphere at infinity, and it often displays interesting geometric and topological features that can reveal algebraic information about the group itself. In this talk, first we will discuss the general theory and present classical examples of limit sets, including some simple fractals (Cantor set, Sierpinski carpet). Then, we will present the construction of groups whose limit set is a Pontryagin sphere. These groups are obtained as reflection groups, and the construction is based on the existence of certain right-angled hyperbolic polyhedra. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.

October 9th
Speaker: Liam Keenan (Brown)
Title: On products of skeleta

Abstract: Given simplicial complexes, X and Y, a classical result of Eilenberg and Zilber tells us that the complex of integral chains on the product, X x Y, is quasi-isomorphic to the tensor product of complexes associated to X and Y. Their result relies on the basic observation that the product of an n-simplex with an m-simplex, can be built by gluing together simplices of dimension (n+m). Remarkably, this basic observation has much farther reaching-consequences than one might expect. In joint work with Maximilien Peroux, we showed that whenever you have two objects, X and Y, built up out of simplicies, the skeletal filtrations of X and Y can always be related to the skeletal filtration of X x Y, in an entirely canonical fashion. I will introduce this circle of ideas, explain my work with Peroux, and discuss its relation with the Dold-Kan correspondence.

October 14th (Tuesday 1:30-2:30pm – cross-listed from the Combinatorics Seminar)
Speaker: Lee Kennard (Syracuse)
Title: Regular Matroids and Torus Representations

Abstract: Recent work with Michael Wiemeler and Burkhard Wilking presents a link between arbitrary finite graphs and torus representations all of whose isotropy groups are connected. The link is via combinatorial objects called regular matroids, which were classified in 1980 by Paul Seymour. We then apply Seymour’s deep result to classify and to compute geometric invariants of this class of torus representations. The applications to geometry are significant. A highlight of our analysis of these representations is the first proof of Hopf’s 1930s Euler Characteristic Positivity Conjecture for metrics invariant under a torus action where the torus rank is independent of the manifold dimension.

October 16th
Speaker: John Abou-Rached (Binghamton)
Title: Integral models for non-Shimura curves and the Eichler-Shimura congruence relation

Abstract: We construct integral models for an infinite family of algebraic curves that includes noncongruence modular curves, as well as curves whose uniformizers are non-arithmetic Fuchsian groups. Most of these curves are not Shimura curves. We affirm a conjecture of Mukamel that the set of primes of good reduction of such curves have arithmetic significance and obtain an explicit description of this set. We conjecture that a version of Deligne-Rapoport's study of the reduction of modular curves holds in this context, and conjecture that a version of the Eichler-Shimura congruence relation holds in this setting, in resonance with Shimura curves.

October 23th
Speaker:
Title:

Abstract:

October 30th
Speaker: Tam Cheetham-West (Yale)
Title: Splittings and finite quotients of 3-manifold groups

Abstract: Embedded essential surfaces in a 3-manifold correspond to non-trivial splittings of its fundamental group. We give some conditions on the fundamental group of a Haken hyperbolic 3-manifold which guarantee that any other 3-manifold group with the same finite quotients must have a non-trivial splitting. Using one of these conditions, we obtain restrictions on the possible first Betti numbers of regular covers of aspherical integer homology spheres. This is joint work with Khánh Lê.

November 6th
Speaker: Genevieve Walsh (Tufts)
Title: Hyperbolic groups vs relatively hyperbolic groups

Abstract: A hyperbolic group is a group that acts geometrically on a hyperbolic metric space, and a relatively hyperbolic group is a group that acts geometrically finitely on a hyperbolic metric space. Sometimes, these spaces are quasi-isometric. This turns out to be exceedingly rare. We will discuss the consequences of such quasi-isometries, and give lots of examples of these types of group actions. This is joint work with Emily Stark.

November 13th
Speaker: Maximilien Peroux (Michigan State University)
Title: Algebraic characterization of cobordism theories

Abstract: Cobordism theories are fundamental invariants in geometry and topology: they organize closed manifolds according to whether or not they bound. They play a central role in classification problems and have deep connections to index theory, characteristic classes, and even modern mathematical physics. Each variant of cobordism is represented in stable homotopy theory by a Thom spectrum, linking geometric questions with homotopical constructions. Recognizing which spectra arise as Thom spectra is a subtle problem, since it amounts to detecting when an abstract homotopy-theoretic object has a geometric origin in cobordism. In ongoing joint work with Brazelton, Calle, Chan, and Keenan, we introduce an algebraic characterization of Thom spectra as certain algebraic objects in stable homotopy theory. This approach provides a new algebraic perspective on geometric phenomena.

November 20th
Speaker: David Chan (Michigan State University)
Title: Building equivariant infinite loop spaces

Abstract: Infinite loop spaces are important kinds of topological spaces which can be used to construct cohomology theories. For a finite group G, there is an equivariant refinement of infinite loop spaces which represent cohomology theories for G-spaces. Infinite loop G-spaces play an important role in recent applications of equivariant homotopy theory, but constructing examples can be difficult. In this talk we will describe some results about how to construct every possible infinite loop G-space from algebraic input.

November 27th
No seminar

December 4th
Speaker: Hongbin Sun (Rutgers)
Title: On virtual chirality of 3-manifolds

Abstract: We prove that if a prime 3-manifold M is not finitely covered by the 3-sphere or a product manifold, then M is virtually chiral, i.e. it has a finite cover that does not admit an orientation-reversing self-homeomorphism. For a non-prime 3-manifold, it is virtually chiral if it has a virtually chiral prime summand. This is joint work with Zhongzi Wang.

Spring 2017

2017-06-08T14:32:05-04:00

Spring 2017

For questions contact Christoforos Neofytidis

January 26
Speaker: Marco Varisco (SUNY at Albany)
Title: Assembly maps for topological cyclic homology

Abstract: I will present recent joint work with Wolfgang Lück, Holger Reich, and John Rognes [arXiv:1607.03557], in which we use assembly maps to study the topological cyclic homology of group algebras. For any finite group G, for any connective ring spectrum A, and for any prime p, we prove that TC(A[G];p) is determined by TC(A[C];p) as C ranges over the cyclic subgroups of G. More precisely, we prove that for any finite group the assembly map with respect to the family of cyclic subgroups induces isomorphisms on all homotopy groups. For infinite groups, we establish pro-isomorphism, split injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map with respect to the family of virtually cyclic subgroups is split injective but in general not surjective—in contrast to what happens in algebraic K-theory.

February 2
Speaker: Matt Zaremsky (Cornell)
Title: Local to global: Discrete Morse theory and topological properties of infinite groups

Abstract: Discrete Morse theory is a tool for turning difficult global problems into easier local ones. For example, one might wish to know whether or not a certain cell complex is connected, simply connected, or n-connected for some higher n, or whether a filtration of a cell complex is homologically stable. Morse theory can reduce these difficult “global” problems to easier “local” questions about the so-called ascending or descending links of vertices. In this talk I will first discuss some background on discrete Morse theory and some historical applications to important questions in geometric group theory, and then describe some of my own contributions.

February 9
Speaker: Wiktor Mogilski (SUNY at Binghamton)
Title: The Strong Atiyah Conjecture and computations of $L^2$-Betti numbers

Abstract:The Strong Atiyah Conjecture predicts that for any group $G$ with bounded torsion, the $L^2$-Betti numbers of any $G$-space are rational, with denominators determined by the order of the torsion subgroups. In this talk we will restrict ourselves to the setting of Coxeter groups, and I will present a special trick that, in many cases, improves the Strong Atiyah Conjecture prediction of the denominators of the $L^2$-Betti numbers. In many examples, this improvement (along with additional work) allows us to make complete computations of the $L^2$-Betti numbers. I will conclude by exploiting this trick to obtain new affirmative results regarding the Singer Conjecture for Coxeter groups. This is joint work with Kevin Schreve.

February 16
Speaker: Tam Nguyen Phan (SUNY at Binghamton)
Title: An analog of the Tits building in nonpositive curvature

Abstract:Locally symmetric manifolds (of noncompact type) form an interesting class of nonpositively curved manifolds. By Borel-Serre, the thin part of the universal cover of an arithmetic locally symmetric space is homotopically equivalent to the rational Tits building, which is homotopically a wedge of spheres of dimension q-1, where Q is the Q-rank of the locally symmetric space. In general, q is less than or equal to n/2. We show that this is not an arithmetic coincidence in a weaker sense, which is that if M is a noncompact, bounded nonpositively curved manifold with finite volume and no arbitrarily small geodesic loops (so that M is tame), then any nontrivial homology cycle in the thin part of \tilde{M} must have dimension less than or equal to n/2 - 1. For each such cycle, we construct a complex at infinity of dimension less than n/2 that is an analog of the Tits building which we collapse the cycle onto. Simplices of such a complex consist of points whose Busemann functions are invariant under a group of parabolic isometries. You don't need to know what the Tits building is but some familiarity with nonpositively curved geometry will be helpful in understanding the talk.

February 23
Speaker: Jason Manning (Cornell University)
Title: k-fold triangle groups

Abstract: I'll introduce a generalization of classical triangle groups and use them to answer a question of Agol and Wise on possible extensions of Wise's Malnormal Special Quotient Theorem (a central tool in the theory of groups acting on cube complexes).

March 2
Speaker: Alex Moody (UT-Austin)
Title: Geography and classification of symplectic fillings of Legendrian surgeries

Abstract: Understanding the smooth topology of 4-manifolds is a notoriously hard problem. Due to a theorem of A.A. Markov, we cannot hope for similar classifications like those in the lower dimensional case. Due to exoticness, we cannot hope to understand 4-manifolds completely just by studying their algebraic topology as we can in the higher dimensional case. However, if we assume a 4-manifold admits a symplectic form and the boundary is a certain contact manifold (with a natural compatibility condition), classification problems suddenly become tractable. We will show how such classification results of Eliashberg, McDuff and Lisca in the case of lens spaces can be used to put topological restrictions (completely determining the Betti numbers and signature) on these symplectic fillings for a very large class of a contact 3-manifolds (infinitely many surgeries on every link in the 3-sphere is a subset). We will then give a conjectural answer about the classification of such fillings.

March 9
Speaker: Caitlin Leverson (Georgia Tech)
Title: Invariants of Legendrian Knots

Abstract: Given a plane field dz-ydx in R^3. A Legendrian knot is a knot which at every point is tangent to the plane at the point. One can similarly define a Legendrian knot in any contact 3-manifold (manifold with a plane field satisfying some conditions). In this talk, we will explore Legendrian knots in R^3, J^1(S^1), and #^k(S^1xS^2) as well as a few Legendrian knot invariants. We will also look at the relationships between a few of these knot invariants. No knowledge of Legendrian knots will be assumed.

March 16
Speaker: Kamlesh Parwani (Eastern Illinois University)
Title: Zero entropy subgroups of the mapping class group

Abstract: Let $M$ be a compact surface with boundary. We are interested in the question of how a group action on $M$ permutes a finite invariant set $X \subset int(M)$. More precisely, how the algebraic properties of the induced group of permutations of a finite invariant set affects the dynamical properties of the group. Our main result shows that in many circumstances if the induced permutation group is not solvable then among the homeomorphisms in the group there must be one with a pseudo-Anosov component. We formulate this in terms of the mapping class group relative to the finite set and show the stronger result that in many circumstances (e.g. if $\partial M \ne \emptyset$) this mapping class group is itself solvable if it has no elements with pseudo-Anosov components. This is joint work with John Franks.

March 28 (Joint with Combinatorics Seminar)
Note special time: 1:15 - 2:15
Speaker: Caroline Klivans (Brown)
Title: On the Connectivity of Three-Fimensional Tilings

Abstract:I will discuss domino tilings of three-dimensional manifolds. I will focus on the connected components of the space of tilings of such manifolds under local moves. Using topological techniques, I introduce two parameters of tilings: the flux and the twist. The main result characterizes, in terms of these two parameters, when two tilings are connected by local moves.

I will not assume any familiarity with the theory of tilings.

March 30
Speaker: Phillip Wesolek (SUNY at Binghamton)
Title: Commensurated subgroups and periodic subgroups of tree almost automorphism groups.

Abstract: (Joint work with A. Le Boudec) The tree almost automorphism groups are non-discrete locally compact completions of the Higman-Thompson groups. The tree almost automorphism groups are independently interesting locally compact groups, and furthermore every group that almost acts on a sufficiently regular rooted tree embeds into one of these groups. We begin by introducing the almost automorphism groups and describing their relationship to the Higman-Thompson groups. We then consider the subgroups such that every element is contained in a compact subgroup; such groups are the topological analogue of torsion subgroups and are called periodic. We show every periodic subgroup is indeed locally elliptic - i.e. every finite set is contained in a compact subgroup. As applications, we recover a result for Thompson's group V as well as a new observation about the Röver group. We finally consider the commensurated subgroups of almost automorphism groups; these subgroups generalize normal subgroups. We show every commensurated closed subgroup of an almost automorphism group is either finite, compact and open, or equal to the entire group.

April 6
seminar cancelled

April 13
Spring break

April 20
Speaker: Teddy Einstein (Cornell)
Title: Hierarchies of Non-Positively Curved Cube Complexes

Abstract: Wise's malnormal special quotient theorem (MSQT) is a key ingredient in Agol's proof of the Virtual Haken Conjecture. The most important step in proving the MSQT is the construction of a hierarchy for hyperbolic compact special non-positively curved cube complexes. In this talk, I will explain what a hierarchy of a compact special non-positively curved cube complex is and discuss how to generalize a new proof of the MSQT by Agol, Groves and Manning to the relatively hyperbolic setting.

April 27
No Seminar, Hilton Lecture
Title:

Abstract:

May 4
Speaker: Ilya Gekhtman (Yale)
Title: Word length asymptotics for actions of some automatic (e.g. relatively hyperbolic) groups.

Abstract:Consider any nonelementary action of a hyperbolic group G on a not necessarily proper Gromov hyperbolic space X. The action is not assumed to be discrete (for example, it could be a dense subgroup of SL_{2}\R) and X is not assumed to be proper (for example it could be the curve complex, on which the mapping class group acts with pseudo-Anosov elements acting as loxodromics).

We prove certain asymptotic properties for the action, including the following. 1)With respect to the Patterson-Sullivan measure on the boundary of G, the image in X of almost every word-geodesic in G sublinearly tracks a geodesic in X. 2)The proportion of elements in a Cayley-ball of radius R in G which act loxodromically on X converges to 1 with R.

A major tool is Cannon's theorem that hyperbolic groups admit geodesic automation. The same result hold for relatively hyperbolic groups with respect to generating sets which admit a geodesic automaton, including geometrically finite Kleinian groups, and more generally to automatic structures satisfying certain axioms related to growth tightness. We also obtain results for more general Markov processes, for example showing a *nonbacktracking* random walk on a group acting nonelementarily on a Gromov hyperbolic space hits loxodromic elements with This is based on completed and ongoing work with Sam Taylor and Giulio Tiozzo.

Spring 2018

2019-02-18T08:49:57-04:00

Spring 2018

January 16 (algebra crosspost - meets in WH-100E at 2:50)
Speaker: Jonas Deré (KU Leuven Kulak)
Title: Which manifolds admit expanding maps

Abstract: In 1981, M. Gromov completed the proof that every manifold admitting an expanding map is, up to finite cover, homeomorphic to a nilmanifold. Since then it was an open question to give an algebraic characterization of the nilmanifolds admitting an expanding map. During my talk, I will start by introducing the basic notions of expanding maps and nilmanifolds. Then I explain how the existence of such an expanding map only depends on the covering Lie group and on the existence of certain gradings on the corresponding Lie algebra. One of the applications is the construction of a nilmanifold admitting an Anosov diffeomorphism but no expanding map, which is the first example of this type.

February 1
Speaker: Jonathan Williams (Binghamton University)
Title: Sewing a homotopy into pieces

Abstract: In this talk, I will try to explain the title. There will be 4-manifolds, many pictures, and very little background needed.

February 8
Speaker: Russell Ricks (Binghamton University)
Title: A Rank Rigidity Result for Certain Nonpositively Curved Spaces via Spherical Geometry

Abstract: To understand the geometry of nonpositively curved (NPC) spaces, it is natural to classify the various types of spaces that can occur. The Rank Rigidity Theorem for compact NPC manifolds separates the class of compact NPC manifolds into three very distinct types, and proves that nothing else can exist.

A version of Rank Rigidity has been conjectured for more general NPC spaces (CAT(0) spaces). In this talk, we discuss some progress toward this general conjecture, by reducing the problem to looking at patterns on spheres. In particular, we prove the conjecture for certain NPC spaces with one-dimensional boundary. Unlike previous results in this area, there are no additional constraints on the CAT(0) space (such as a manifold or polyhedral structure).

February 15
Speaker: Carlos Vega (Binghamton University)
Title: Null Distance on a Spacetime

Abstract: In contrast to the Riemannian setting, a Lorentzian manifold (M,g) is not known to possess any naturally induced distance function. I will first try to explain why that is, starting with some of the basics of spacetime (Lorentzian) geometry. We will then discuss a `null distance function' introduced in joint work with Christina Sormani, some of its properties, examples, and some open questions.

February 22 (special two-part talk: see next entry)
Speaker: Olakunle Abawonse (Binghamton University)
Title: Topology of the Grunbaum-Hadwiger-Ramos Hyperplane Mass Partition Problem

Abstract: In this talk, we will discuss a problem raised by Ramos that asks for the smallest dimension $d=\Delta(j,k)$ such that for any $j$ measures in $\mathbb{R}^d$, there are $k$ affine hyperplanes that simultaneously cut each measure into $2k$ equal parts. We will give a general configuration space/test map scheme for this problem and show how the theory of relative equivariant obstruction theory applies to this problem.

This is part of a candidacy talk, with committee Laura Anderson (chair), Ross Geoghegan and Michael Dobbins. It is open to all.

February 22 (Special time: 4:15pm)
Speaker: Olakunle Abawonse (Binghamton University)
Title: Hyperplane Mass Partitions Via Relative Equivariant Obstruction Theory

Abstract: We will give solutions to some of the few cases in which the minimum value $d=\Delta(j,k)$ is known. We will achieve this by showing the non-existence of a certain $G$-equivariant map. By the theory of relative equivariant obstruction theory, this problem reduces to evaluating some obstruction classes.

This is part of a candidacy talk, with committee Laura Anderson (chair), Ross Geoghegan and Michael Dobbins. It is open to all.

March 1
Speaker: Jun Li (University of Michigan)
Title: The symplectomorphism groups of rational surfaces

Abstract: This talk is on the topology of $Symp(M,\omega)$, where $Symp(M,\omega)$ is the symplectomorphism group of a symplectic rational surface $(M,\omega)$. We will illustrate our approach with the 5 point blowup of the projective plane. For an arbitrary symplectic form on this rational surface, we are able to determine the symplectic mapping class group (SMC) and describe the answer in terms of the Dynkin diagram of Lagrangian sphere classes. In particular, when deforming the symplectic form, the SMC of a rational surface behaves in the way of forgetting strand map of braid groups. We are also able to compute the fundamental group of $Symp(M, \omega)$ for an open region of the symplectic cone. This is a joint work with Tian-Jun Li and Weiwei Wu.

March 8
Speaker: Lisa Piccirillo (UT-Austin)
Title: Stein Knot Traces

Abstract: Four-manifolds which admit a Stein structure have many nice properties, for example the Stein structure gives bounds on the genus function of the manifold and Stein cobordisms induce nontrivial maps on the Heegaard Floer homology of the boundary. However, handed an arbitrary four-manifold it can be difficult to determine whether it admits a Stein structure. A question in the field asked whether it is ever straightforward to detect Stein structures on particularly simple manifolds; more technically it asked whether the four manifold $X_n(K)$ obtained by attaching an $n$-framed 2-handle to $B^4$ along $K$ is Stein if and only if $n<\overline{tb}(K)$. We answer this in the negative, and in fact show that a Stein $X_n(K)$ can have $n$ arbitrarily much larger than $\overline{tb}(K)$. This talk will focus on the constructive part of our proof, a technique due largely to Osoinach for building knots $K$ and $K’$ with $X_n(K)$ diffeomorphic to $X_n(K’)$. This is joint work with Tom Mark and Faramarz Vafaee.

March 15 (Special time 2:30 pm)
Speaker: Gili Golan (Vanderbilt University)
Title: Invariable generation of Thompson groups

Abstract: A subset $S$ of a group $G$ invariably generates $G$ if for every choice of $g(s)\in G$, $s\in S$ the set $\{s^{g(s)}:s\in S\}$ is a generating set of $G$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this talk, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

March 22
Speaker: Yair Hartman (Northwestern University)
Title: Stationary C*-Dynamical Systems

Abstract: We introduce the notion of stationary actions in the context of C*-algebras, and prove a new characterization of C*-simplicity in terms of unique stationarity. This ergodic-theoretic characterization provides an intrinsic understanding for the relation between C*-simplicity and the unique trace property, and provides a framework in which C*-simplicity and random walks interact. Joint work with Mehrdad Kalantar.

March 29
Speaker: Yuhang Liu (Penn)
Title: Closed 6-manifolds with Positive Sectional Curvature and Non-Abelian symmetry

Abstract: Understanding the structure of Riemannian manifolds with strictly positive sectional curvature remains a fundamental problem in Riemannian geometry. In this talk, I will briefly go over the history of the classification of positively curved manifolds in low dimensions under certain symmetry assumptions on the isometry group. Then I will focus on dim 6 and discuss positively curved 6-manifolds whose isometry groups are non-Abelian Lie groups. Examples of such manifolds will be given together with the isometric group actions, and if time permits I will present some results I got in this direction. This is ongoing work on my thesis problem.

April 12 2018 Hilton Memorial lecture
Speaker: Vaughan Jones (Vanderbilt)
Title: Local scale transformations in one dimension

Special time and place: 3pm–4pm in AA G023

Abstract: In two dimensional conformal field theory, local scaling symmetry means invariance of some kind under conformal transformations. The quantum theory splits into two one dimensional theories called the “chiral halves”. Conformal invariance then gives a projective representation of the the diffeomorphism group (of the line or the circle) on each of the chiral halves. In an attempt to approximate this local scaling invariance we have considered the Thompson groups F and T as approximations to the diffeomorphism groups. Though this does not work perfectly, it has yielded a kind of “topsy turvy” version of chiral CFT including an interesting family of unitary representations of F and T whose coefficients give, among other things, a way to construct all knots and links from elements of F and T, analogous to the standard construction from the braid groups.

April 12
Speaker: Vaughan Jones (Vanderbilt)
Title: The Wysiwyg representations of the Thompson groups

This is a topology seminar talk in WH-100E at a special time, 1:15pm. See the entry immediately below for the Hilton lecture, immediately following this seminar.

Abstract: I will describe a general construction of actions of Thompson’s groups F, T and V and focus on a special kind - unitary representations on a (necessarily infinite dimensional) Hilbert space, coming from very simple combinatorial data. They can be approached via their matrix coefficients which are literally visible. There are many open questions but at least for one family of combinatorial data we can decide equivalence and irreducibility.

April 19
Speaker: Akram Alishahi (Columbia)
Title: Khovanov homology and unknotting number

Abstract: Khovanov homology is a combinatorially-defined knot invariant which refines the Jones polynomial. In this talk we will recall the definition of Khovanov homology and one of its refined versions called Bar-Natan homology, and we will show that the order of h-torsion classes in Bar-Natan homology gives a lower bound for unknotting number.

Special date and time: April 24, 1:15pm in WH-100E (joint with combinatorics)
Speaker: Robert Connelly (Cornell)
Title: Tensegrities: Geometric Structures Suspended in Midair

Abstract: Suppose you have a finite collection of points in Euclidean space or the plane. Some pairs are connected by inextendible cables, others by incompressible struts, and some by fixed length bars. The artist Kenneth Snelson constructed several large structures, made of cables and bars, that hold their shape under tension, where the struts appear to be suspended in midair. Buckminster Fuller, the architect and inventor, called them “tensegrities” because of their “tensional integrity”. But why do they hold their shape? There is a very simple principle using quadratic energy functions that provides the key to their stability. I will show a catalog of highly symmetric tensegrities, created with the help of a little bit of representation theory, as well as tangible models, where you can feel their rigidity first-hand.

Special date and time: May 1, 1:15pm in WH-100E (joint with combinatorics)
Speaker: Boris Bukh (Carnegie Mellon)
Title: Topological Version of Pach's Overlap Theorem

Abstract: Consider the collection of all the simplices spanned by some n-point set in $\mathbb{R}^d$. There are several results showing that simplices defined in this way must overlap very much. In this talk I focus on the generalization of these results to 'curvy' simplices.

Specifically, Pach showed that every $d+1$ sets of points $Q_1, \ldots, Q_{d+1}$ in $\mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. In joint work with Alfredo Hubard, we show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size $C(\log n)^{1/(d-1)}$. We show that this is tight in dimension 2, for all surfaces other than $S^2$. Surprisingly, the optimal bound for $S^2$ is $(\log n)^{1/2}$. This improves upon results of Bárány, Meshulam, Nevo, and Tancer.

May 3
Speaker: Jamie Conway (UC Berkeley)
Title: Classifying Contact Structures on Hyperbolic 3-Manifolds

Abstract: Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but notably not on any hyperbolic manifolds. In this talk, I will discuss a new classification result on an infinite family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot. This is joint work with Hyunki Min.

Spring 2019

2019-05-10T11:31:46-04:00

Spring 2019

February 14
Speaker: Steve Ferry (Rutgers and Binghamton)
Title: How big is epsilon?

Abstract (PDF): A 1979 theorem of Chapman-Ferry says that if $M$ is a compact connected topological $n$-manifold* without boundary with topological metric $d$, then there is an $\epsilon > 0$ so that if $f:M \to N$ is a map from $M$ to a connected manifold of the same dimension such that $diam(f^{-1}(x))<\epsilon$ for every $x \in N$, then $f$ is homotopic to a homeomorphism.

This theorem and its descendants play a continuing role in the work of Farrell-Jones, Bartels-Lück, and others on the Novikov, Borel, and Farrell-Jones Conjectures, the general strategy being to apply ideas from dynamics to “squeeze” a given homotopy equivalence to an appropriately “controlled” equivalence to which some version of the theorem quoted above applies.

We will show that the behavior of $\epsilon$ in our old theorem depends on results from algebraic topology on the vanishing of the $K$-homology of Eilenberg-MacLane spaces of torsion groups. An application to computational topology is suggested.

This is joint work with Alexander Dranishnikov and Shmuel Weinberger.

*Chapman-Ferry did the cases $n\geq5$. The case $n=4$ is due to Freedman-Quinn and $n=3$ follows from work of Perelman.

February 21
Speaker: Ben Dozier (Stony Brook)
Title: Equidistribution of billiard trajectories and translation surfaces

Abstract: Consider a billiard ball bouncing around on a polygonal table. This dynamical system is surprisingly complex. When the angles of the table are rational, the billiard table can be “unfolded” to get a closed surface with a natural flat geometry. This is an example of a translation surface. Billiard trajectories on the original table unfold to straight lines on the translation surface. Translation surfaces form a moduli space (which is a bundle over $M_g$, the moduli space of genus g Riemann surfaces), and this space comes equipped with a natural action by $SL_2(\mathbb R)$. Through a technique of “renormalization”, questions about the dynamics on a fixed surface can be translated into questions about the dynamics associated with this $SL_2(\mathbb R)$ action. The $SL_2(\mathbb R)$ action is very rich, and analogies with homogeneous dynamics can be leveraged.

February 28
Speaker: Cary Malkiewich (Binghamton)
Title: Parametrized spectra and fixed-point theory

Abstract: In 1980, Dold and Puppe presented a revisionist proof of the Lefschetz fixed point theorem. The main idea is that the Lefschetz number L(f) is secretly more than a number, it's actually a map of spectra. Their ideas can be generalized to the Reidemeister trace R(f), or to families of fixed-point problems, but these generalizations require us to work with parametrized spectra, in other words spectra that vary over a fixed base B. I'll talk about what these words mean, and some cool things we can prove once we have them in our toolbox.

March 7
Speaker: Yash Lodha (Ecole Polytechnique federale de Lausanne)
Title: Finitely generated simple left orderable groups, commutator width and orderable monsters

Abstract: In 1980 Rhemtulla asked whether there exist finitely generated simple left orderable groups. In joint work with Hyde, we construct continuum many such examples, thereby resolving this question. In recent joint work with Hyde, Navas, and Rivas, we demonstrate that among these examples are also so called ``left orderable monsters”. This means that all their actions on the real line are of a certain desirable dynamical type. This resolves a question from Navas's 2018 ICM proceedings article concerning the existence of such groups.

March 14
Speaker: Mayank Goswami (Queens College, CUNY)
Title: Computing Extremal Quasiconformal Mappings

Abstract: By the Riemann mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally (i.e., in an angle preserving manner). However, when this map is extended to the boundary it need not necessarily map the vertices of P to those of Q. For many applications it is important to find the “best” vertex-preserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the so-called dilatation) over all points in P. Teichmuller (1940) proved the existence and uniqueness of such maps, which are called extremal quasiconformal maps, or Teichmuller maps. There are many efficient ways to compute or approximate conformal maps, and a result by Bishop computes a (1+ϵ)-approximation of the Riemann map in linear time. However, there is currently no such algorithm for extremal quasiconformal maps (which generalize conformal maps), and only heuristics have been studied so far.

We solve the open problem of finding a finite time procedure for approximating Teichmuller maps in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly (in O(poly(1/ϵ)) iterations) to a (1+ϵ)-approximation of the Teichmuller map, and in the limit to the exact Teichmuller map. Furthermore, every step of the iteration involves convex optimization and solving differential equations, two operations which may be solved in polynomial time in a discrete implementation. Our method uses a reduction of the polygon mapping problem to the punctured sphere problem, thus solving a more general problem. Building upon our results in the continuous setting, we give a discrete algorithm for computing Teichmuller maps.

SPECIAL DATE AND TIME: March 26, 1:15pm in WH 100E
Speaker: Laura Anderson (Binghamton)
Title: Homotopy Types of Combinatorial Grassmannians

Abstract: See the combinatorics page (this is a cross-listing).

April 4 - Peter Hilton Memorial Lecture
Special time and place: LH 9, 3pm
Speaker: Shmuel Weinberger (University of Chicago)
Title: How hard is algebraic topology? Between the constructive and the non.

Abstract: In algebraic topology one studies geometric problems and problems of constructing and deforming highly nonlinear functions by means of algebra. If one knows that two maps are homotopic (i.e. can be deformed to one another) because a certain calculation says they both lie in the trivial group, then what has one learned? (A striking example of this is Smale's turning the sphere inside out, which now can be seen after much highly nontrivial effort, on youtube.) The question I shall discuss is how hard is it to understand what the algebraic topologists tell us.

April 11
Speaker: Matt Zaremsky (Albany)
Title: Bestvina-Brady Morse theory on Vietoris-Rips complexes

Abstract: Discrete Morse theory is a powerful tool for leveraging “local” topological information about a cell complex to make “global” topological conclusions. One prominent incarnation is the version developed by Bestvina and Brady, which has proven invaluable in the study of topological finiteness properties of groups. In this talk I will discuss a generalization of Bestvina-Brady Morse theory that is tailor-made for analyzing Vietoris-Rips complexes of certain metric spaces. I will also discuss some applications to topological data analysis and geometric group theory.

April 18
Speaker: Elizabeth Field (UIUC)
Title: Trees, dendrites, and the Cannon-Thurston map

Abstract: When $1\to H\to G\to Q\to 1$ is a short exact sequence of three word-hyperbolic groups, Mahan Mj has shown that the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point $z$ in the Gromov boundary of $Q$ an ``ending lamination'' on $H$ which consists of pairs of distinct points in the boundary of $H$. We prove that for each such $z$, the quotient of the Gromov boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free group and $Q$ is a convex cocompact purely atoroidal subgroup of $Out(F_N)$, one can identify the resultant quotient space with a certain $R$-tree in the boundary of Culler-Vogtmann's Outer space.

April 25
Speaker: Hyunki Min (Georgia Tech)
Title: Contact structures on hyperbolic manifolds

Abstract: There are two basic questions in contact topology: Which manifolds admit tight contact structures, and on those that do, can we classify tight contact structures? There have been a lot of studies for many prime manifolds, especially for Seifert fibrations and toroidal manifolds. In this talk, we present such a classification on an infinite family of hyperbolic 3-manifolds. This is a joint work with James Conway.

May 2
Speaker: Kate Ponto (University of Kentucky)
Title: Defining additive invariants

Abstract: I'll talk about first steps in an approach to defining additive invariants that captures familiar invariants and many of their most useful properties. This approach defines the character of a representation and the Reidemeister trace of an endomorphism of a compact manifold. It creates a framework that describes both induction and restriction formulas for characters and the compatibility of fixed point invariants with maps of subcomplexes and fibrations.

May 9
Speaker: Jacob Russell-Madonia (CUNY)
Title: The geometry of subgroup combination theorems

Abstract: While producing subgroups of a group by specifying generators is easy, understanding the structure of such a subgroup is notoriously difficult problem. In the case of hyperbolic groups, Gitik utilized a local to global property for geodesics to produce an elegant condition which ensures a subgroup generated by two elements (or more generally generated by two subgroups) will split as an amalgamated free product over the intersection of the generators. We show that a large class of groups demonstrate a similar local to global property from which an analogy of Gitik's result can be obtained. This allows for a generalization of Gitik's theorem in many important classes of groups including CAT(0) groups, the mapping class groups of a surface, and 3-manifold groups. Joint work with Davide Spriano and Hung C. Tran.

Spring 2020

2020-08-08T08:34:09-04:00

Spring 2020

February 13
Speaker: Cary Malkiewich (Binghamton University)
Title: The equivariant parametrized $h$-cobordism theorem

Abstract: The classical $h$-cobordism theorem plays a critical role in the classification of high-dimensional smooth manifolds up to diffeomorphism. The “parametrized” or “higher” $h$-cobordism theorem of Waldhausen gives access to the rest of the diffeomorphism group $Diff(M)$ in a stable range, using algebraic $K$-theory. I will describe recent and ongoing work with Tom Goodwillie and Mona Merling on the $G$-equivariant form of Waldhausen's theorem when $G$ is a finite group.

February 20
Speaker: Michael Ching (Amherst College)
Title: Tangent ∞-categories and Goodwillie calculus

Abstract: The theory of Goodwillie calculus uses an analogy between homotopy theory and differential geometry to make systematic decompositions of homotopy-theoretic functors into “polynomial” pieces. For example, a suitable functor (say from the category of topological spaces to itself) has a “Taylor tower”, a sequence of polynomial approximations that, in good cases, can be used to recover information about the original functor.

Cockett and Cruttwell (following Rosický) have developed an abstract framework which axiomatizes the categorical properties of the tangent bundle functor on the category of smooth manifolds, and includes other “tangent bundle” constructions in areas such as algebraic geometry and synthetic differential geometry, among others.

In this talk I will describe joint work with Kristine Bauer and Matthew Burke that puts Goodwillie calculus into this same “tangent category” framework (or, rather, its ∞-categorical counterpart) and thus makes precise the hitherto informal analogy developed by Goodwillie. I will argue, in particular, that the Taylor tower construction can be recovered in a formal way from this underlying tangent structure. This work sets the scene for importing other concepts from differential geometry, such as connections and curvature, into homotopy theory.

February 27
Speaker: James Hyde (Cornell University)
Title: Sufficient conditions for a group of homeomorphisms of the Cantor set to be 2-generated

Abstract: We say a group G of homeomorphisms of the Cantor set is vigorous if for any non-empty clopen set A and non-empty proper clopen subsets B and C of A there exists g in G wholly supported on A with Bg a subset of C. This talk represents joint work with Collin Bleak and Luke Elliott. We will also discuss some properties of vigorous simple groups of homeomorphisms of the Cantor set. In particular finitely generated vigorous simple groups of homeomorphisms of the Cantor set are 2-generated.

March 5

No seminar (Winter break)

March 12
Speaker: John Klein (Wayne State University)
Title: Hypercurrents

Abstract: This talk poses the question as to what a higher dimensional analog of a continuous time Markov chain might be, in which the time parameter is replaced by arbitrary smooth manifold.

As a partial answer, we introduce the notion of a “protocol,” which consists of a space whose points are labeled by real numbers indexed by the set of cells of a fixed CW complex in prescribed degrees, where the labels are required to vary continuously. When the space is a one-dimensional manifold, then a protocol determines a continuous time Markov process.

In the presence of a homological gap condition, we associate to each protocol a ‘characteristic’ cohomology class which we call the hypercurrent. The hypercurrent comes in two flavors: one algebraic topological and the other analytical. For generic protocols we show that the analytical hypercurrent tends to the topological hypercurrent in the 'low temperature' limit.

We also exhibit examples of protocols having nontrivial hypercurrent.

The remaining talks were cancelled due to the coronavirus outbreak.

March 19
Speaker: Kathryn Mann (Cornell University)
Title: TBA

March 26
Speaker: John Lind (California State University Chico)
Title: TBA

April 23
Speaker: Vitaly Lorman (University of Rochester)
Title: TBA

PETER HILTON MEMORIAL LECTURE
SPECIAL TIME AND LOCATION: April 30, 3pm, LH009
Speaker: Robert Gompf (University of Texas at Austin)
Title: Exotic Smooth Structures on $\mathbb R^4$

Abstract: One of the most surprising discoveries in 4-manifold topology was the existence of smooth manifolds homeomorphic, but not diffeomorphic, to Euclidean 4-space. For fundamental reasons, this phenomenon can only occur in 4 dimensions. We will survey the subject, from its origin to recent developments regarding symmetries of such manifolds.

May 7
Speaker: Marco Varisco (SUNY Albany)
Title: TBA

Spring 2021

2021-08-26T16:28:16-04:00

Spring 2021

February 25
Speaker: Christopher Perez (Loyola University New Orleans)
Title: Towers and elementary embeddings in toral relatively hyperbolic groups

Abstract: In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if H is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group G, then G is a tower over H. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.

March 4
Speaker: Kevin Schreve (University of Chicago)
Title: Generalized Tits Conjecture for Artin groups

Abstract: In 2001, Crisp and Paris showed the squares of the standard generators of an Artin group generate an “obvious” right-angled Artin subgroup. This resolved an earlier conjecture of Tits. I will introduce a generalization of this conjecture, where we ask that a larger set of elements generates another “obvious” right-angled Artin subgroup. I will give evidence that this is a good generalization, explain what classes of Artin groups we can prove it for, and give some applications. All of it is joint work with Kasia Jankiewicz.

March 25
Speaker: Alexander Margolis (Vanderbilt University)
Title: Topological completions of quasi-actions and discretisable spaces

Abstract: A fundamental problem in geometric group theory is the study of quasi-actions. We introduce and investigate discretisable spaces: spaces for which every cobounded quasi-action can be quasi-conjugated to an isometric action on a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups are discretisable spaces, but the property holds much more generally. For instance, most hyperbolic groups are discretisable, as are most finitely presented groups of cohomological dimension two.

Along the way, we introduce the concept of the topological completion of a quasi-action. This is a locally compact group, well-defined up to a compact normal subgroup, reflecting the geometry of the quasi-action. We give several applications of the tools we develop. For instance, we show that any finitely generated group quasi-isometric to a ‬Z‭-by-hyperbolic group is also Z-by-hyperbolic.

April 1
Speaker: Olakunle Abawonse (Binghamton University)
Title: Gelfand and MacPherson's combinatorial formula for Pontrjagin classes, part I: the topology

Abstract: Let $X$ be a simplicial manifold. A smoothing of $X$ is a smooth manifold M together with a homeomorphism from X to M that is smooth on each closed simplex. Rohlin and \v{S}varc(1957) and Thom(1958) showed that all smoothings of X have the same rational Pontrjagin classes. This raised the hope for a combinatorial formula for these classes. In 1992 Gelfand and MacPheron announced such a formula and gave a very terse proof. This is the first of two talks. The second will be given by Laura Anderson in the Combinatorics Seminar (on 4/6). In these two talks we'll explain their proof. The first part of their proof is an alternative form of Chern-Weil theory, which will be the topic of Part 1.

April 15
Speaker: Nate Fisher (Tufts University)
Title: Boundaries, random walks, and nilpotent groups

Abstract: In this talk, we will discuss boundaries and random walks in the Heisenberg group. We will discuss a class of sub-Finsler metrics on the Heisenberg group which arise as the asymptotic cones of word metrics on the integer Heisenberg group and describe new results on the boundaries of these polygonal sub-Finsler metrics. After that, we will explore experimental work to examine the asymptotic behavior of random walks in this group. Parts of this work are joint with Sebastiano Nicolussi Golo.

April 22
Speaker: Emily Stark (Wesleyan University)
Title: Action Rigidity for Graphs of Manifold Groups

Abstract: The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically.

A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we prove certain graphs of manifold groups are action rigid. Consequently, we obtain examples of quasi-isometric groups that do not virtually have a common model geometry. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

April 29
Speaker: Uylsses Alvarez (Binghamton University)
Title: Order complexes and tropical phased matroids

Abstract: A topological poset is a Hausdorff space with a partial ordering such that the relation is closed in the product space. An interesting feature of topological posets is that they can be associated to a generalization of the order complex of discrete posets. In this talk, I will mainly focus on my favorite example of topological posets: the tropical phase hyperfield. More specifically I will give the homeomorphism type of the order complex of the covector set of a tropical phased matroid.

May 6
Speaker: Wouter van Limbeek (University of Illinois – Chicago)
Title: Do thin groups have discrete commensurators?

Abstract: Let G be a simple Lie group and \Gamma < G a lattice. In 1974, Margulis proved that if the commensurator of \Gamma is dense, then \Gamma is arithmetic. In 2015, Shalom asked if the same is true only assuming \Gamma is Zariski-dense in G. I will report on recent progress on this question for normal subgroups of lattices in rank 1 (e.g. hyperbolic space) using ideas from infinite ergodic theory, Brownian motion, random walks and harmonic maps. I will attempt to give a picture of how all these ideas combine to give information on commensurators. This is joint work with D. Fisher and M. Mj.

May 13
Speaker: Achim Krause (Muenster)
Title: Witt vectors with coefficients and characteristic polynomials over non-commutative rings

Abstract: Witt vectors are classically discussed as a means to canonically lift characteristic p objects to mixed characteristic. These “p-typical” Witt vectors also have an analogue that combines all primes at once, the “big Witt vectors”. These show up naturally in the study of refinements of topological Hochschild homology, as TR_0. Since the latter makes sense more generally for noncommutative rings, and even with coefficients in a bimodule, it is natural to ask for a similar generalisation of Witt vectors on the algebraic side. We describe these ``Witt vectors with coefficients'' algebraically, and show that they enjoy analogs of a lot of the usual structure of Witt vectors. We also see how in this perspective, the trace from cyclic K-theory can be interpreted as a kind of noncommutative characteristic polynomial.

Spring 2022

2022-07-19T12:41:10-04:00

Spring 2022

February 3rd
Speaker: Sarah Blackwell (U Georgia) (online talk)
Title: Triple Knot Grid Diagrams

Abstract: In this talk I will introduce a project I have been working on which uses trisections of 4-manfiolds to represent ``Lagrangian-like’’ surfaces in $\mathbb{CP}^2$ by ``triple knot grid diagrams.’’ Gay and Kirby defined a decomposition of (smooth, closed, connected, oriented) 4-manifolds called a trisection, and proved that every such 4-manifold admits this decomposition. Meier and Zupan showed that surfaces embedded in 4-manifolds inherit a trisection from the trisection of the 4-manifold. Their work includes a description of how to represent these surfaces with ``shadow diagrams.’’ In this project I consider specific shadow diagrams of surfaces in $\mathbb{CP}^2$ that naturally arise as grid diagrams on the central surface of the standard (genus one) trisection of $\mathbb{CP}^2$. The result is a process for encoding Lagrangian-like surfaces which appear to be combinatorial representations of Lagrangian surfaces in $\mathbb{CP}^2$. Surprisingly, triple knot grid diagrams representing Lagrangian-like surfaces are sparse; adding the extra necessary condition makes such diagrams hard to find.

February 10th
Speaker: Matt Zaremsky (SUNY-Albany) (online talk)
Title: Higher virtual algebraic fibering of certain right-angled Coxeter groups

Abstract: A group is said to “virtually algebraically fiber” if it has a finite index subgroup admitting a map onto Z with finitely generated kernel. Stronger than finite generation, if the kernel is even of type F_n for some n then we say the group “virtually algebraically F_n-fibers”. Right-angled Coxeter groups (RACGs) are a class of groups for which the question of virtual algebraic F_n-fibering is of great interest. In joint work with Eduard Schesler, we introduce a new probabilistic criterion for the defining flag complex that ensures a RACG virtually algebraically F_n-fibers. This expands on work of Jankiewicz–Norin–Wise, who developed a way of applying Bestvina–Brady Morse theory to the Davis complex of a RACG to deduce virtual algebraic fibering. We apply our criterion to the special case where the defining flag complex comes from a certain family of finite buildings, and establish virtual algebraic F_n-fibering for such RACGs. The bulk of the work involves proving that a “random” (in some sense) subcomplex of such a building is highly connected, which is interesting in its own right.

February 17th
Speaker: Rachel Skipper (Ohio State) (online talk)
Title: Finiteness properties for braided and ribboned groups of homeomorphisms of the Cantor Set

Abstract: The braided Higman-Thompson groups were first introduced independently by Brin and Dehornoy. In this talk, we talk about some generalizations of this construction as well as how to braid self-similar groups. The focus of the talk will be on some recent work about finiteness properties of the resulting groups and how they fit into the growing field of big mapping class groups.

February 24th
Speaker: Oğuz Şavk (Boğaziçi University) (online talk)
Title: Homology 3-spheres bounding contractible 4-manifolds and homology 4-balls

Abstract: A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results. Our approach is based on Mazur’s famous argument and its generalization, they together provide a unification of all recognized results.

March 3rd
Speaker: Cary Malkiewich (Binghamton)
Title: Counting periodic orbits of flows

Abstract: In this informal talk I'll describe work in progress on a new method for counting periodic orbits of flows, in an “algebraic” or “homological” way. The main breakthrough in recent months is the construction of a new invariant, using spectra, that simultaneously generalizes two earlier constructions by Fuller and by Geoghegan and Nicas. The computation of this invariant is work in progress, but I'll give a few examples where we can do the computation.

March 10th
Speaker: Gabriel Islambouli (UC Davis) (online talk)
Title: Stable equivalence of smooth 4-manifolds described as sequences of handlebodies

Abstract: Following constructions of numerous authors, one can build a smooth 4-manifold from a loop of Morse functions on a surface, a loop in the cut complex, a loop in the pants complex, or from a multisection diagram. In this talk, we will discuss these constructions, as well as outline a stable equivalence theorem for these descriptions so that, for example, any two loops of Morse functions on a surface corresponding to the same smooth 4-manifold are related by a sequence of given operations.

March 24th
Speaker: David Mehrle (Cornell) (in person talk)
Title: Towards an equivariant Hochschild—Kostant—Rosenberg Theorem

Abstract: The Hochschild—Kostant—Rosenberg (HKR) theorem is a bridge between algebra and geometry with applications in algebraic K-theory, Lie theory, deformation theory, and other fields. For a smooth $k$-algebra $A$, the HKR theorem gives an isomorphism between the Kähler differentials of $A$ (a geometric object) and the Hochschild homology of $A$ (an algebraic gadget). We conjecture that, when a finite group $G$ acts on $A$ by ring homomorphisms, the HKR theorem becomes a $G$-equivariant isomorphism. In this talk, I will share some progress towards proving this conjecture, and discuss some of the obstacles that remain. This is joint work-in-progress with J.D. Quigley and Michael Stahlhauer.

March 31st
Speaker: Steven Gindi (Binghamton)
Title: Long Time Limits of Generalized Ricci Flow

Abstract: We derive rigidity results for generalized Ricci flow blowdown limits on classes of nilpotent principal bundles. We accomplish this by constructing new functionals over the base manifold that are monotone along the flow. This overcomes a major hurdle in the nonabelian theory where the expected Perleman-type functionals were not monotone and did not yield results. Our functionals were inspired and built from subsolutions of the heat equation, which we discovered using the nilpotency of the structure group and the flow equations. We also use these and other new subsolutions to prove that, given initial data, the flow exists on the principal bundle for all positive time and satisfies type III decay bounds.

April 7th
Speaker: Sam Shepherd (Vanderbilt)
Title: Semistability of cubulated groups

Abstract: I will discuss my theorem that cubulated groups are semistable at infinity, together with background on these two concepts. I will also present a result about modifying the cubulation of a group to achieve certain geometric features, which is needed to prove the semistability theorem.

April 14th
Speaker: Patrick Naylor (Princeton)
Title: Doubles of Gluck twists

Abstract: The Gluck twist of an embedded 2-sphere in the 4-sphere is a 4-manifold that is homeomorphic, but not obviously diffeomorphic to the 4-sphere. Despite considerable study, these homotopy spheres have resisted standardization except in special cases. In this talk, I will discuss some conditions that imply the double of a Gluck twist is standard, i.e., is diffeomorphic to the 4-sphere. This is based on joint work with Dave Gabai and Hannah Schwartz.

April 21st
Speaker: Chaitanya Tappu (Cornell)
Title: Mapping Class Group acts continuously on the marked moduli space

Abstract: We define a moduli space of marked hyperbolic structures on an infinite type surface, analogous to Teichmueller spaces. The mapping class group of the surface acts on this marked moduli space. For infinite type surfaces, the mapping class group is a topological group, so we can ask if the above action is continuous. We answer this in the affirmative.

April 28th
Speaker: Matt Durham (UC Riverside)
Title: Local quasicubicality and sublinear Morse geodesics in mapping class groups and Teichmuller space

Abstract: Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinear-Morse boundary to more broadly capture these generic directions.

In joint work with Abdul Zalloum, we develop the geometric foundations of sublinear-Morseness in the mapping class group and Teichmuller space. We prove that their sublinearly-Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear-Morseness in terms of the hierarchical structure on these spaces.

Our techniques include developing tools for modeling sublinearly-Morse rays via CAT(0) cube complexes. Part of this analysis involves establishing a direct connection between the geometry of the curve graph and the combinatorics of hyperplanes in these cubical models.

May 5th
Speaker: Yash Lodha (University of Vienna)
Title: Some new constructions in the theory of left orderable groups

Abstract: : I will define two new constructions of finitely generated simple left orderable groups (in recent joint work with Hyde and Rivas). Among these examples are the first examples of finitely generated simple left orderable groups that admit a minimal action by homeomorphisms on the Torus, and the first family that admits such an action on the circle. I shall also present examples of finitely generated simple left orderable groups that are uniformly simple (these were constructed by me with Hyde in 2019). And present new examples that, somewhat surprisingly, have infinite commutator width. Finally, I will present some new results around the second bounded cohomology of these groups (joint with Fournier-Facio).

Spring 2023

2023-08-08T12:42:40-04:00

Spring 2023

January 19th
Speaker: Marco Varisco (SUNY Albany)
Title: Universal Spaces for Proper Actions, Vietoris–Rips Complexes, and Equivariant Discrete Morse Theory

Abstract: I will talk about a joint article with Matt Zaremsky (now at Albany and formerly a postdoc at Binghamton, like me). Our main result is that all asymptotically CAT(0) groups have finite universal spaces for proper actions, given by Vietoris–Rips complexes. My goal is to introduce all these concepts and explain why we care about them. The technique we develop to prove this result is an equivariant version of Bestvina–Brady discrete Morse theory, which we believe to be of independent interest.

https://doi.org/10.1112/blms.12534

January 26th
Speaker: Bastiaan Cnossen (U Bonn) Zoom talk
Title: Traces and categorification

Abstract: The trace of a linear operator is simple to define, yet appears all over mathematics in many disguises: from characters of representations, through fixed-point formulas, to various geometric transfer maps. The theory of oo-categories and higher algebra allows one to organize many of these occurrences of the trace within a formal unified calculus. This calculus is more intricate and elaborate than one might expect, because some of its fundamental features are revealed only by categorification, leading to investigations of traces in $(\infty,2)$-categories.

In this talk, I will describe joint work with Shachar Carmeli, Maxime Ramzi and Lior Yanovski that sets up a general “character theory” for studying, among other things, the interaction of traces with colimits by an “induced character formula” (generalizing and refining work of Ponto-Shulman). The interaction between traces and categorification plays a key role in our approach. I will also explain how this theory can be applied to the study of the Becker-Gottlieb transfer and of topological Hochschild homology of Thom spectra.

February 2nd
No seminar this week

February 9th
Speaker: Carissa Slone (Rochester)
Title: Two-slices over cyclic groups of prime power order

Abstract: The slice filtration focuses on producing certain spectra, called slices, from a genuine G-spectrum X over a finite group G. We have a complete characterization of all 1-, 0-, and (-1)-slices for any such G, and a characterization for 2-slices over $C_2$ and Klein-4. We will characterize 2-slices over $C_p$ ($p$ odd) and expand this characterization to $C_{p^n}$.

February 16th
Speaker: Patricia Cahn (Smith College)
Title: Trisected 4-Manifolds as Branched Covers of the 4-Sphere

Abstract: Trisections of 4-manifolds, introduced by Gay and Kirby as a 4-dimensional analog of Heegaard splittings in dimension 3, are a powerful mechanism for importing techniques from 3-dimensional topology into dimension 4. A branched cover of the 4-sphere, equipped with its standard trisection, along a (possibly singular) surface in bridge position, gives rise to a trisected 4-manifold. A natural question is which trisected 4-manifolds arise this way, and for those that do, what can be said about the degree of the cover or complexity of the branching set. We discuss this problem for the case of geometrically simply-connected 4-manifolds, joint with Blair, Kjuchukova and Meier. If time permits, we will discuss algorithms for computing invariants of the trisected covering manifold in terms of the corresponding permutation representation of the group of the branching set, joint with Alishahi, Matic, Pinzón-Caicedo, and Ruppik.

February 23rd
Speaker: Didac Martinez-Granado (UC-Davis)
Title: Two notions of duality for geodesic currents

Abstract: Geodesic currents are a suitable closure of the space of curves on a hyperbolic surface introduced by Bonahon in 1986. Notions such as the geometric intersection number of curves extend to geodesic currents. I will discuss two equivalent viewpoints on geodesic currents: as dual curve functionals and as dual spaces. On the one hand, a geodesic current induces a functional on the space of curves on the surface via intersection number with the current. We say that such a curve functional is ``dual to the current”. In joint work with Dylan Thurston, we give sufficient and necessary conditions for curve functionals to be dual to geodesic currents. On the other hand, a geodesic current together with a choice of hyperbolic metric induces a Gromov hyperbolic space, that we call a ``dual space of the current”. In joint work with Luca De Rosa, we describe the metric structure of such spaces.

March 2nd
No seminar this week

March 9th
Speaker: Daniel Gulbrandsen (University of Wisonsin Milwaukee)
Title: Cubical Collapses and a New Compactification of Locally-Finite CAT(0) Cube Complexes

Abstract: In this talk we will define what it means for a cube complex to be collapsible. In particular, our definition will apply to the case that the complex is not finite. Then, we will show that all locally-finite CAT(0) cube complexes are collapsible. The process will yield an inverse sequence of finite convex subcomplexes whose inverse limit provides a Z-compactification of the complex in which the boundary (which we call the cubical boundary) incorporates properties of both the visual and Roller boundaries.

March 16th
No seminar this week

March 23rd
Speaker: Thomas Koberda (U Virginia)
Title: The first order theory of homeomorphism groups of compact manifolds

Abstract: I will describe some recent work on the first order theory of homeomorphism groups of manifolds. I will discuss a new result which shows that the homeomorphism groups of two compact manifolds are elementarily equivalent if and only if the two manifolds are homeomorphic, which resolves an old conjecture of Rubin. I will then describe some of the expressive power of the language of groups in the theory of homeomorphism groups, with implications for the subgroup structure of homeomorphism groups, and for the descriptive set theory of these groups.

March 30th
Speaker: Justin Barhite (U Kentucky)
Title: Traces and Cotraces in Bicategories

Abstract: Traces arise in many different places in math: traces of matrices, characters of group representations, and even the Euler characteristic of a CW complex! There are very general notions of trace, expressed in the language of category theory, that capture these examples of traces and whose properties imply familiar results like the Lefschetz fixed point theorem and the induction formula for characters. The formalism of traces doesn't tell the whole story though; there are some constructions that feel trace-like in certain ways but also have a distinct flavor, and what's really needed to explain them from this category-theoretic perspective is a dual notion of “cotrace.” I will talk about some of these things that I have been working to understand by developing a theory of bicategorical cotraces.

April 6th
No seminar this week (spring break)

April 13th
Speaker: Maru Sarazola (Johns Hopkins)
Title: Fibrant transfer for model structures

Abstract: Model structures are robust categorical structures that provide an abstract framework to do homotopy theory. Unfortunately, in practice it is often very hard to prove that something satisfies the requirements of a model structure. To this end, there are several results in the literature that explore techniques for constructing model structures on a given category. Of particular note is the Transfer theorem, allowing the user to transfer a model structure along an adjunction.

After a review of model structures, this talk will present a new generalization of the transfer theorem where the relevant homotopical structure is only transferred between fibrant objects. Time permitting, we will explore some applications. Based on recent work with Leonard Guetta, Lyne Moser and Paula Verdugo.

April 20th
Speaker: Benjamin Thompson (Cornell)
Title: Khovanov homology of rational tangles

Abstract: Links are fundamental objects of study in low-dimensional topology, and are generalized by tangles. One of the most well-studied link invariants, the Jones polynomial, was generalized by Khovanov to a homology theory in the late 90s, before being extended to tangles in 2004 by Bar-Natan. Working with tangles instead of links in Khovanov homology can greatly simplify calculations and more easily shed light on the underlying properties of the theory. In this talk we examine a subclass of tangles known as rational tangles, and show that their Bar-Natan homology can be computed with a string replacement algorithm. This essentially makes computing Khovanov homology for rational knots trivial, and provides an elementary explanation of the resulting homological phenomena.

April 27th
Speaker: Sahana Balasubramanya (SUNY Buffalo)
Title: Actions of solvable groups on hyperbolic spaces

Abstract: (joint work with A.Rasmussen and C.Abbott) Recent papers of Balasubramanya and Abbott-Rasmussen have classified the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace-Cornulier-Monod-Tessera. This machinery applies in particular to solvable groups with virtually cyclic abelianizations.

May 4th
Speaker: Alexander Kupers (U Toronto)
Title: Pontryagin classes of Euclidean space bundles

Abstract: Tangent bundles of smooth manifolds are vector bundles, and tangent bundles of topological manifolds are Euclidean space bundles. Characteristic classes of vector bundles, like the Pontryagin classes, also make sense to Euclidean space bundles, but surprisingly behave very differently. I will explain recent joint work with Manuel Krannich on this topic, with a focus on the relationship to diffeomorphism groups of discs.

Spring 2024

2024-08-07T10:14:53-04:00

Spring 2024

February 15th
Speaker: Sebastian Hurtado (Yale University)
Title: Groups with full Limit Set vs Lattices

Abstract: We show the existence of an (infinitely generated) discrete subgroup of a Lie group (such as SL_n(R)) that has full limit set in its (Furstenberg) boundary and which is not a lattice, we also discuss the possibility of whether this is possible for finitely generated groups. All notions will be explained. Based on work in progress with Subhadip Dey.

February 22nd
Speaker: Nima Hoda (Cornell University)
Title: Tree of graph boundaries of hyperbolic groups

Abstract: Regular trees of graphs are inverse limits of particularly simple inverse systems of finite graphs. They form a 1-dimensional subclass of the Markov compacta: a class of finitely describable inverse limits of simplicial complexes, which includes all boundaries of hyperbolic groups. I will discuss upcoming joint work with Jacek Swiatkowski in which we use Bowditch's canonical JSJ decomposition to characterize the 1-ended hyperbolic groups whose boundaries are (regular) trees of graphs.

February 29th
Speaker: Xin Li (University of Glasgow) (virtual talk)
Title: Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

Abstract: Topological groupoids describe orbit structures of dynamical systems by capturing their local symmetries. The group of global symmetries, which are pieced together from local ones, is called the topological full group. This construction gives rise to new examples of groups with very interesting properties, solving outstanding open problems in group theory. This talk is about a new connection between groupoids and topological full groups on the one hand and algebraic K-theory spectra and infinite loop spaces on the other hand. Several applications will be discussed. Parts of this connection already feature in work of Szymik and Wahl on the homology of Higman-Thompson groups.

March 7th
No seminar this week (spring break)

March 14th
Speaker: Lucas Williams (Binghamton University)
Title: Periodic Points and Equivariant Parameterized Cobordism

Abstract: In this talk we investigate invariants that count periodic points of a map. Given a self map $f$ of a compact manifold we could detect $n$-periodic points of $f$ by computing the Reidemeister trace of $f^n$ or by computing the equivariant Fuller trace. In 2020 Malkiewich and Ponto showed that the collection of Reidemeister traces of $f^k$ for varying $k|n$ and the equivariant Fuller trace are equivalent as periodic point invariants, and they conjecture that for families of endomorphisms the Fuller trace will be a strictly richer invariant for $n$-periodic points.

In this talk we will explain our new result which confirms Malkiewich and Ponto's conjecture. We do so by proving a new Pontryagin-Thom isomorphism between equivariant parameterized cobordism and the spectrum of sections of a particular parametrized spectrum and using this result to carry out geometric computations.

March 21st
Speaker: Lei Chen (University of Maryland)
Title: Mapping class groups of circle bundles over a surface

Abstract: In this talk, we study the algebraic structure of mapping class group Mod(M) of 3-manifolds M that fiber as a circle bundle over a surface. We prove an exact sequence, relate this to the Birman exact sequence, and determine when this sequence splits. We will also discuss the Nielsen realization problem for such manifolds and give a partial answer. This is joint work with Bena Tshishiku and Alina Beaini.

March 28th
Speaker: Mark Pengitore (University of Virginia)
Title: Residual finiteness growth functions of the mapping class group and the question of linearity

Abstract: Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. One potential application of these functions is towards linearity of the mapping class group, and we will present some partial progress towards understanding these functions for the mapping class group.

April 4th
Speaker: Giuseppe Martone (Sam Houston State University)
Title: Correlation theorem and (cusped) Hitchin representations

Abstract: Given distinct hyperbolic structures m and m' on a closed orientable surface, how many closed curves have m- and m'-length roughly equal to x, as x gets large? Schwartz and Sharp's correlation theorem answers this question. Their explicit asymptotic formula involves a term exp(Mx) and 0<M<1 is the correlation number of the hyperbolic structures m and m'.

In this talk, we will show that the correlation number can decay to zero as we vary m and m', answering a question of Schwartz and Sharp. Then, we discuss extensions of this correlation theorem to the context of higher rank Teichmüller theory and find diverging sequences of SL(3,R)-Hitchin representations along which the correlation number stays uniformly bounded away from zero.

This talk is based on joint work with Xian Dai and joint work in progress with Nyima Kao.

April 11th
PETER HILTON MEMORIAL LECTURE
SPECIAL TIME AND LOCATION: April 11, 3pm, Lecture Hall 009
Speaker: Alex Eskin (University of Chicago)
Title: Polygonal Billiards and Dynamics on Moduli Spaces

Abstract: Billiards in polygons can exhibit bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry, Teichmuller theory and ergodic theory on homogeneous spaces. The talk will be an introduction to these ideas, aimed at a general mathematical audience.

April 18th
Speaker: Alex Wright (University of Michigan)
Title: Spheres in the curve graph and linear connectivity of the Gromov boundary

Abstract: For a vertex $c$ and an integer radius $r$, the sphere $S_r(c)$ is the induced graph on the set of vertices of distance $r$ from $c$. We will show that spheres in the curve graph are typically connected, and discuss connectivity properties of the Gromov boundary. We will also explain the motivation and context for this work, touching tangentially on Cannon's conjecture and convex cocompactness.

April 25th
No seminar this week (Monday classes meet)

May 2nd
Speaker: Cary Malkiewich (Binghamton University)
Title: A Solomon-Tits theorem for arbitrary hyperplane collections

Abstract: Suppose we take an arbitrary collection of hyperplanes in n-dimensional Euclidean, hyperbolic, or spherical geometry, along with all of their nonempty intersections. These form a partially ordered set, so we can take the realization and get a topological space, called the Tits complex. One version of the Solomon-Tits theorem says that, if we were to take *all* hyperplanes, the space we get is homotopy equivalent to a wedge of spheres of dimension (n-1).

In this talk I'll describe how to prove a variant of this theorem where we can take just about any reasonable subset of the hyperplanes, and the result still holds. We can furthermore give a presentation of the homology of the resulting space: it has a generator for each polytope cut out by the hyperplanes, and the relations encode subdivision of the polytopes. The proof is quite fun, it's an inductive proof where we add the hyperplanes one at a time and count how many new polytopes, and spheres in the Tits complex, are created. Our main application is to the groups of cut-and-paste operations between these polytopes.

Spring 2025

2025-08-22T11:45:19-04:00

Spring 2025

January 23rd
Organizational meeting, meet in WH 100E at 2:50pm

January 30th
Speaker: Matthew Zaremsky (University at Albany)
Title: Progress around the Boone-Higman conjecture

Abstract: The Boone-Higman conjecture (1973) predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. There has been a flurry of recent activity around this conjecture, in particular relating it to the family of so called twisted Brin-Thompson groups. In this talk I will give some background on the conjecture, give a gentle introduction to twisted Brin-Thompson groups, and then discuss various recent results of mine, including some joint with combinations of Jim Belk, Collin Bleak, Francesco Fournier-Facio, James Hyde, and Francesco Matucci.

February 6th
Problem session

February 13th
No seminar

February 20th
No seminar

February 27th
Speaker: Valentina Zapata Castro (University of Virginia)
Title: Monoidal complete Segal spaces

Abstract: Viewing a monoid as a category with a single object allows us to encode the binary operation using the properties of composition and associativity inherent in any category. In this talk, we use this idea to explore the relationship between $(\infty,1)$-categories with a monoidal structure and $(\infty,2)$-categories with one object. This exploration relies on the model structure of simplicial and $\Theta_2$-spaces. The talk is designed to be self-contained, requiring no prior knowledge of the aforementioned categories.

March 6th
Speaker: Inhyeok Choi (Cornell University)
Title: Genericity of pseudo-Anosovs and quasi-isometries

Abstract: In this talk, I will explain a recent result that pseudo-Anosov mapping classes are generic in every Cayley graph of mapping class groups. If time permits, I will also explain why this strategy goes well with quasi-isometries and implies genericity of Morse elements for groups quasi-isometric to (many) 3-manifold groups and special cubical groups.

March 13th
Spring break

March 20th
Peter Hilton Memorial Lecture

March 27th
Speaker: Colby Kelln (Cornell University)
Title: Coning off a hyperbolic manifold with totally geodesic boundary

Abstract: Let $M$ be a compact hyperbolic manifold with totally geodesic boundary. If the injectivity radius of $\partial M$ is larger than an explicit function of the normal injectivity radius of $\partial M$, we show there is a negatively curved metric on the space obtained by coning each boundary component of $M$ to a point. Moreover, we give explicit geometric conditions under which a locally convex subset of $M$ gives rise to a locally convex subset of the cone-off. Group-theoretically, we conclude that the fundamental group of the cone-off is hyperbolic and the $\pi_1$-image of the locally convex subset is a quasi-convex subgroup.

April 3rd
Speaker: Theodore Weisman (University of Michigan)
Title: Anosov representations of cubulated hyperbolic groups

Abstract: An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which imitates the behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, which shows that every hyperbolic group that acts geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. Mainly, the proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.

April 10th
Speaker: Marco Volpe (University of Toronto)
Title: Fiberwise simple homotopy theory

Abstract: Simple homotopy theory is, roughly speaking, the study of finite CW-complexes up to collapses and expansions. From its early stages, it has been observed that simple homotopy types are deeply connected to K-theory. This connection is realized through Wall's finiteness obstruction for finitely dominated complexes and the Whitehead torsion of a homotopy equivalence between finite complexes. One of Waldhausen's main contributions ('83) to simple homotopy theory was to incorporate both Wall's obstruction and the Whitehead torsion in the study of assembly maps in K-theory. Later on, Dwyer-Weiss-Williams ('03) have introduced “fiberwise” assembly maps associated to fibrations over a fixed base space, thereby providing a framework for understanding simple homotopy types varying in families.

In this talk, we introduce a novel perspective on fiberwise assembly maps, developed via the infinity-category of sheaves of spectra on a topological space. Using this approach, we are able to simultaneously generalize both the recently announced (but as yet unpublished) work of Bartels-Efimov-Nikolaus and the topological Dwyer-Weiss-Williams index theorem ('03).

This is a joint work with Maxime Ramzi and Sebastian Wolf.

April 17th
Speaker: Sayantika Mondal (CUNY)
Title: Distinguishing filling curves via designer metrics

Abstract: There are many topological invariants one can associate with homotopy classes of closed curves. These include algebraic and geometric self-intersection number, intersection with curves in a class of curves (for example, simple ones), the Goldman bracket, complementary component types of a curve, mapping class group stabilizers of a curve, and many others. How these invariants interact and determine the curve type (mapping class group orbit) is an active area of research today. In this talk, we focus on the so called inf invariant (shortest length metric) associated to a filling curve, its relationship with the geometric self-intersection number, and its relation to the optimal metric that is tailored to produce the minimum length. While clearly the geometric self-intersection number is a type invariant, we address whether the inf invariant can distinguish between curves that have the same self-intersection. This is joint work with Ara Basmajian.

April 24th
Speaker: Kasia Jankiewicz (UC Santa Cruz / IAS)
Title: Cubical quotients of cubical nonproduct groups

Abstract: Burger-Mozes constructed examples of simple groups acting geometrically on a CAT(0) complex, which is a product of trees. As a counterpoint, we prove that every group acting geometrically on a CAT(0) cube complex which is not a product, admits a nontrivial quotient which also admits a geometric action on a CAT(0) cube complex. Our construction relies on the cubical version of small cancelation theory. This is joint work with M. Arenas and D. Wise.

Tuesday April 29th, 1:15-2:15 pm (joint with the Combinatorics Seminar)
Speaker: Leo Jiang (Toronto)
Title: Topology of Real Matroid Schubert Varieties

Abstract: Every linear representation of a matroid determines a matroid Schubert variety whose geometry encodes combinatorics of the matroid. When the representation is over the real numbers, we study the topology of the real points of the variety. Our main tool is an explicit cell decomposition, which depends only on the oriented matroid structure and can be extended to define a combinatorially interesting topological space for any oriented matroid. This is joint work with Yu Li.

May 1st - DOUBLE HEADER
Speaker: Chaitanya Tappu (Cornell University) – 2:50-3:50pm
Title: Contractibility of the marked moduli space

Abstract: We prove that the marked moduli space of any infinite type surface is contractible. The marked moduli space of an infinite type surface (equipped with an action of the big mapping class group) is introduced as the generalisation of the usual Teichmüller space of a finite type surface. This result is analogous to that of Douady–Earle, who proved that the (quasiconformal) Teichmüller space of an arbitrary Riemann surface, whether of finite or infinite type, is contractible. Even though the marked moduli space reduces to the Teichmüller space in case the surface is of finite type, it is quite distinct from the Teichmüller space in case the surface is of infinite type. Nevertheless, we are able to adapt the Douady–Earle proof to the setting of the marked moduli space. A key difference is that in this setting, we use a Fréchet space topology on the vector space of (-1, 1)-forms (that is, Beltrami forms), rather than the usual Banach space topology.

Speaker: Filippo Calderoni (Rutgers) – 4:15 - 5:15pm
Title: Groups, orders, and dynamics: a new perspective

Abstract: A countable group G is said to be left-orderable if it admits a total order which is invariant under left multiplication, or, equivalently, if G admits a faithful action by orientation preserving homeomorphisms on the real line. There is a beautiful connection between the algebraic properties of a left-orderable group G and the conjugacy action on LO(G), the compact Hausdorff space of all left-orders supported by G. In this talk I will survey some results towards characterizing those left-orderable groups such that the orbit space of LO(G) modulo conjugacy is trivial from the viewpoint of descriptive set theory.