seminars:topsem

## Geometry and Topology Seminar

Unless stated otherwise, the seminar takes place Thursdays at 2:50–3:50 pm in WH-100E followed by refreshments served from 4:00–4:25 pm in WH-102.

Some seminar speakers will also give a colloquium talk at 4:30 pm on the same day as the seminar talk. This seminar is partly funded as one of Dean's Speaker Series in Harpur College (College of Arts and Sciences) at Binghamton University.

To receive announcements of seminar talks by email, please join the seminar's mailing list. To subscribe to an on-line calendar with the seminar schedule, please choose a format: iCal or xml.

### 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:

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• March 23
Speaker: Ramón Vera (Penn State)
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• March 30
Speaker: Phillip Wesolek (SUNY at Binghamton)
Title: The structure of simple totally disconnected locally compact groups via embeddings with dense image

Abstract: (Joint work with P.-E. Caprace and C. Reid) The collection of topologically simple totally disconnected locally compact (t.d.l.c.) groups which are compactly generated and non-discrete forms a rich and compelling class of locally compact groups, denoted by S; members include the simple algebraic groups over non-archimedean local fields and the almost automorphism groups. In recent years, a general theory for these groups, which considers the interaction between the geometry and the topology, has emerged. In this talk, we study this interaction by considering locally compact groups H which embed densely into some group G in S. We show the structure of such groups H is very restrictive as soon as H is non-discrete. As applications, we show no group in S admits a compact open subgroup with an infinite solvable Sylow subgroup. We further place restrictions on the automorphism groups of groups in S.

• April 6
Speaker: Adam Saltz (University of Georgia)
Title: TBA

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• April 13
Spring break
• April 20
Speaker: Teddy Einstein (Cornell)
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• April 27
No Seminar, Hilton Lecture
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• May 4
Speaker: Ilya Gekhtman (Yale)
Title: TBA

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