Department of Mathematics and Statistics, Binghamton University

The Arithmetic Seminar

2026-04-10T16:24:03-04:00

The Arithmetic Seminar

TOPICS: Arithmetic in the broadest sense that includes Number Theory (Elementary Arithmetic, Algebraic, Analytic, Combinatorial, etc.), Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Geometry of Numbers, Tropical Geometry, Arithmetic Dynamics, Arithmetic Topology, etc.

PLACE and TIME: This semester the seminar meets primarily on Tuesdays at 4:00 pm, with possible special lectures at other days and times. The in-house talks will be in-person, while visitors outside of Binghamton area will be in-person or by Zoom: Zoom link

ORGANIZERS:
Regular Faculy: Alexander Borisov, Marcin Mazur, Adrian Vasiu.
Post-Docs: Huy Dang

Current Ph.D. students: Hari Asokan, Mithun Padinhare Veettil.

Graduated Ph.D. students (in number theory and related topics): Ilir Snopce (Dec. 2009), Xiao Xiao (May 2011), Jinghao Li (May 2015), Ding Ding (Dec. 2015), Patrick Milano (May 2018), Changwei Zhou (May 2019), Patrick Carney (May 2023), Sarah Lamoureux (Sep. 2023), Sayak Sengupta (May 2024).

SEMINAR ANNOUNCEMENTS: To receive announcements of seminar talks by email, please join our mailing list.

Related seminar: Upstate New York Online Number Theory Colloquium (online, irregular): http://people.math.binghamton.edu/borisov/UpstateNYOnline/Colloquium.html

Previous Arithmetic Seminar Talks

————————- Fall 2025
Spring 2025 ——– Fall 2024
Spring 2024 ——– Fall 2023
Spring 2023 ——– Fall 2022
Spring 2022 ——– Fall 2021
Spring 2021 ——– Fall 2020
Spring 2020 ——– Fall 2019
Spring 2019 ——– Fall 2018
Spring 2018 ——– Fall 2017
Spring 2017 ——– Fall 2016
Spring 2016 ——– Fall 2015
Spring 2015 ——– Fall 2014

Spring 2026

January 27
Speaker: NA
Title: Organizational Meeting
Abstract:

February 3 (2:45-3:45 pm, cross-listed from Algebra Seminar)
Speaker: Tim Riley (Cornell University)
Title: Conjugator length
Abstract: The conjugacy problem for a finitely generated group $G$ asks for an algorithm which, on input a pair of words u and v, declares whether or not they represent conjugate elements of $G$. The conjugator length function $CL$ is its most direct quantification: $CL(n)$ is the minimal $N$ such that if $u$ and $v$ represent conjugate elements of $G$ and the sum of their lengths is at most $n$, then there is a word $w$ of length at most $N$ such that $uw=wv$ in $G$. I will talk about why this function is interesting and how it can behave, and I will highlight some open questions. En route I will talk about results variously with Martin Bridson, Conan Gillis, and Andrew Sale, as well as recent advances by Conan Gillis and Francis Wagner.

February 10
Speaker: Alexander Borisov (Binghamton)
Title: A structure sheaf for Kirch topology, an update
Abstract: Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. I will give an update on my current work on a natural presheaf of functions on this topological space: locally integer polynomial functions. In particular, I will discuss when the sheafification is equal to the presheaf, and when it is bigger. I will also discuss (Cech) cohomology. In particular, I will give examples with trivial and nontrivial H^1. No prior knowledge of the topic is assumed. This talk will also serve as an introduction to Mithun's talk next week.

February 17
Speaker: Mithun Veettil (Binghamton)
Title: Some results on the Locally LIP functions
Abstract: Locally LIP functions are obtained as a result of sheafification of the presheaf LIP on some infinite subset $X$ of $N={1,2,3,\ldots}$, with a prescribed topology. Often we work with Kirch topology on $N$ that makes $N$ a connected, locally connected, and Hausdorff space.
If the set $X$ is a union of non-connected open sets, then we can easily define a locally LIP function on $X$ that is not a LIP function globally. In fact, even if the space $X$ is connected, a locally LIP function on $X$ need not be a LIP function on $X$. In this talk, we will look at $X=N$\ $6N$, which is connected, and construct a locally LIP function that is not LIP on $X$. Also, we will show that this is not the case if one works with $\mathbb{Z}[1/2]$ instead of $\mathbb{Z}$ for the above set $X$.

February 24
Speaker: Hari Asokan (Binghamton)
Title: Variation of Geometric Invariant Theory
Abstract: Geometric Invariant Theory is used to construct quotients of group actions on varieties, but the outcome depends on a choice of linearization. Variation of Geometric Invariant Theory (VGIT) studies the different quotients resulting from changing this choice. In this talk I will give an informal introduction to VGIT, focusing on how stability changes as linearization varies.

March 3
Speaker: Connor Stewart (CUNY)
Title: Conductor–Discriminant Inequality for Tamely Ramified Cyclic Covers
Abstract: We consider $\mathbb{Z}/n$-covers $X\to\mathbb{P}^1$ defined over discretely valued fields $K$ with excellent valuation ring $\mathcal{O}_K$ and perfect residue field of characteristic not dividing $n$. Two standard measures of bad reduction for such a curve $X$ are the Artin conductor of its minimal regular model over $\mathcal{O}_K$ and the valuation of the discriminant of a Weierstrass equation for $X$. We prove an inequality relating these two measures. Specifically, if $X$ is given by an affine equation $y^n = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ is bounded above by $(n-1)v_K(\textrm{disc (rad}\ f))$. This extends previous work of Ogg, Saito, Liu, Srinivisan, and Obus-Srinivasan on elliptic and hyperelliptic curves. (Joint work with Andrew Obus and Padmavathi Srinivasan.)

March 10
Speaker: Eric Yin (Binghamton)
Title: Generating abelian extensions with elliptic curves with complex multiplication.
Abstract: An elliptic curve with complex multiplication is one with extra endomorphisms, ones that are not simply given by multiplication-by-m maps. In this talk we discuss how this extra structure allows us to find an analogy to Kronecker-Weber, generating abelian extensions of imaginary quadratic fields through torsion points on elliptic curves with CM. In addition, we discuss the role the ideal class group plays in both measuring ramification and classifying elliptic curves with a given endomorphism ring. This lets us describe both the Hilbert class field, the maximal unramified abelian extension, and the maximal abelian extension of any imaginary quadratic field.

March 17
Speaker: Anitha Srinivasan (Comillas University, Madrid), by Zoom
Title: The generalized Markoff equation
Abstract: The talk will look at various aspects of the generalized Markoff equation $a^2+b^2+c^2=3abc+m$ ($m\ge 0$), giving an overview of all the exciting work in the area. A few examples of topics that will be mentioned are: the classification of solution triples $(a, b, c)$ that come from $k$-Fibonacci sequences, open conjectures (which $m's$ have no solutions?), counting algorithms for the number of solutions (trees) and the Markoff equation mod $p$.

March 24
Speaker: Jauing Jun (SUNY New Paltz)
Title: Categorical approach to stability of tropical toric vector bundles
Abstract: Developing a suitable notion of vector bundles in tropical geometry has recently attracted considerable interest. In a recent work, Khan and Maclagan introduced tropical vector bundles using matroids, inspired by Klyachko’s classification of toric vector bundles, and studied their stability properties. In this talk, we reinterpret their notion of stability through the framework of the categorical approach to stability proposed by André. This perspective clarifies the structure underlying their results and places them in a broader conceptual setting. This is joint work with Alex Sistko and Cameron Wright.

April 14
Speaker: Anubhav Nanavaty (Cornell University)
Title: Feynman Integrals and Symmetric Matrices
Abstract: Since the foundational work of Broadhurst and Kreimer, there has been a significant push to understand the amplitudes of Feynman Integrals as periods, or integrals of algebraic functions over algebraic domains. Mysteriously, some of these amplitudes are special values of the Riemann zeta function (and more generally multiple zeta values). Work of Brown suggests that these amplitudes are related to the homology of Kontesevich's graph complex, and therefore the cohomology of the moduli space of curves by work of Chan, Galatius and Payne. Central to the story are the Borel classes - GLn(Z) equivariant cohomology classes on the space of projective symmetric matrices of full rank. I first show that, in a very general setting, the Voevodsky motive of this space splits into a direct sum of Tate Motives. I conclude by work in-progress with collaborators, where we aim to compute the weights of the Borel classes viewed as differential forms on the space of projective complex symmetric matrices of full rank and use this computation to express certain classes of Feynman amplitudes to multiple zeta values.

April 21
Speaker: Joe Kramer-Miller (Lehigh University)
Title: TBA
Abstract: TBA

April 28
Speaker: TBA
Title: TBA
Abstract: TBA

Geometry and Topology Seminar - Kuijper TA

2026-04-10T10:15:06-04:00

Geometry and Topology Seminar

We meet Thursdays at 2:45–3:45 pm in Whitney Hall 100E. This semester's organizers are James Hyde and Lorenzo Ruffoni. The seminar has an announcement mailing list open to all.

Topics include: geometric group theory, differential geometry and topology, low-dimensional topology, algebraic topology, and homotopy theory.

Spring 2026

January 29h
Speaker: Juliet Aygun (Cornell)
Title: Counting geodesics on prime-order k-differentials

Abstract: It has been of popular interest over the last several decades to count geodesics with respect to their length on flat surfaces. Asymptotics of these counting functions for generic translation surfaces, which are Riemann surfaces with a holomorphic one form, have been determined by the pioneering work of Eskin-Masur-Zorich. There is a more general type of flat surface called a (1/k)-translation surface, which is a Riemann surface with a k-differential. Equivalently, a (1/k)-translation surface is a collection of polygons in the complex plane with sides identified pairwise by translation and possible rotations of 2pi/k. In this talk, we will discuss asymptotics of these counting functions on generic (1/k)-translation surfaces when k is prime and genus is more than two. The main tools I will discuss are GL+(2,R)-orbit closures and a result of Eskin-Mirzakhani-Mohammadi which relates asymptotics to GL+(2,R)-orbit closures.

February 5th
Speaker: Barry Minemyer (Commonwealth University - Bloomsburg)
Title: Negatively Curved Metrics on Complex Hyperbolic Branched Covers

Abstract: Gromov and Thurston used hyperbolic branched cover manifolds to construct the first known examples of compact manifolds which admit a pinched negatively curved metric, but do not admit a hyperbolic metric. Fine and Premoselli (n=4) and Hamenstadt and Jackel (n > 4) later used these same manifolds to construct the first known examples of negatively curved Einstein metrics (in these respective dimensions) that are not locally symmetric.

Recently, Stover and Toledo proved that analogous complex hyperbolic branched cover manifolds exist. They also proved that these manifolds do not admit a locally symmetric metric, and a result of Zheng shows that these manifolds are Kahler. In this talk I will present recent work proving the existence of pinched negatively curved metrics, as well as the existence of negatively curved Kahler-Einstein metrics (due to Guenancia and Hamenstadt) on these complex hyperbolic branched cover manifolds. Part of my presented work is joint with Lafont.

February 12th
Speaker: Francis Wagner (Cornell University)
Title: Isoperimetric functions and the Word Problem

Abstract: A fundamental algorithmic question in group theory is the Word Problem for finitely generated groups, which asks whether there exists an algorithm to decide whether two words on the generators represent the same group element. A related notion is the Dehn function of a finitely presented group, the smallest isoperimetric function of the presentation's Cayley complex. While the Dehn function gives an upper bound for the complexity of the Word Problem for that group, this bound is only meaningful in the class of finitely presented groups and is very far from sharp even in this class. We resolve this disconnect by instead considering the Dehn functions of the finitely presented groups into which a group embeds, demonstrating a refinement of the Higman embedding theorem that gives a potentially quasi-optimal bound on the Dehn function of the ambient group.

February 19th
Speaker: Satya Howladar (University of Florida)
Title: Gromov’s Conjecture for Product of Baumslag-Solitar Groups and some other One relator groups

Abstract: Gromov introduced macroscopic dimension of metric spaces in order to study large scale properties of manifolds. He conjectured that a closed $n$-manifold which admits Positive Scaler Curvature metric, should have its universal cover to be of macroscopic dimension at most $n-2$, with respect to the pull back metric on it. This conjecture depends a lot on the fundamental group of the base manifold. For $n>4$, closed spin $n$-manifolds $M$, we developed sufficient condition on $\pi_1(M)$, to verify the conjecture. When $\pi_1(M)$ is product of $2$-dimensional groups (i.e. groups with classifying space a $2$-dimensional CW complex), $\mathbb Z_2$-summands in their homology creates a problem for application of our technique. We could resolve this in the case of certain one-relator groups, including Baumslag-Solitar, and certain others, by passing to some finite index subgroup of them not admitting $\mathbb Z_2$-torsion in homology. This is done by the well-known technique of Fox calculus, to analyze boundary maps of cells of finite index covers. I will try to revisit this technique and sketch a proof our result.

February 26th
Speaker: Lucas Williams (Binghamton University)
Title: Equivariant Framed 1-Manifolds and the Pontryagin-Thom Isomorphism

Abstract: The Pontryagin-Thom construction gives an isomorphism between the cobordism group of framed n-manifolds and the nth stable homotopy group of the sphere spectrum. The G-equivariant Pontryagin-Thom construction gives an isomorphism between the cobordism group of V-framed G-manifolds and the Vth stable homotopy group of the G-equivariant sphere spectrum. We will discuss both of these constructions and then present some new explicit descriptions of the images of each 1-dimensional manifold equipped with an action by the cyclic group of order 2 in their relevant homotopy groups. We subsequently provide a new perspective on some key differences between the equivariant and non-equivariant Hopf fibration.

March 5th
Speaker: Oliver Wang (University of Virginia)
Title: Group actions on exotic spheres

Abstract: In the 1960's W.C. Hsiang and W.Y. Hsiang showed that exotic spheres admit less symmetries than the standard sphere. However, constructing symmetries on exotic spheres has been a difficult task. It is still an open question whether or not every exotic sphere admits a smooth, nontrivial S^1-action. In fact, it is open whether or not every exotic sphere admits a smooth, nontrivial C_p-action, where C_p denotes the cyclic group of order p. In this talk, I will discuss recent work with Nick Kuhn and J.D. Quigley relating this problem to stable homotopy theory.

March 12th
Speaker: Sofia Martinez (Bryn Mawr College)
Title: Localizations on Lattices

Abstract: Homotopical Combinatorics is a newer area of Algebraic Topology that studies in a more tractable manner the homotopical structure of topological spaces with the action of a group. The main objects of study in this new area are Transfer systems, originally created with the goal of understanding equivariant analogs of higher coherences. In a more category-theoretic language, a transfer system on a poset, or more generally a finite category, C, is a wide subcategory of C closed under pullbacks. This talk will focus on the case when C=Sub(G), the subgroup lattice of a finite group, G. Subsequent work shows that transfer systems occur naturally as the acyclic fibrations of nicer categories known as model category and their are universal constructions that provide a way to move between these categories; these are called left and right Bousfield localizations. In this we will see how transfer systems change under these types of constructions.

March 13th
PETER HILTON MEMORIAL LECTURE
SPECIAL TIME AND LOCATION: March 13, 3:30pm, Alumni Lounge at Old O'Connor Hall
Speaker: Martin Bridson (University of Oxford)
Title: Chasing finite shadows of infinite groups through geometry

Abstract: There are many situations in geometry or elsewhere in mathematics where it is natural or convenient to explore infinite groups of symmetries via their actions on finite objects. But how hard is it find these finite manifestations and to what extent does the collection of all such actions determine the infinite group?

In this colloquium, I will sketch some of the rich history of such problems and then describe some of the great advances in recent years. I'll describe pairs of distinct groups that have the same finite images and I'll sketch the proof of some “profinite rigidity results”, i.e. theorems showing that in certain circumstances one can identify an infinite group if one knows its set of finite images.

March 19th
Speaker: Francesco Lin (Columbia)
Title: Coexact 1-form spectral gaps of hyperbolic rational homology spheres

Abstract: The spectral gap of the Hodge Laplacian of functions (or, equivalently, exact 1-forms) is a very well-studied fundamental quantity associated to a hyperbolic three-manifold. In recent years, the problem of understanding its counterpart on coexact 1-forms has also spurred a lot of activity because of its relation with questions in number theory and low-dimensional topology. In this talk, after introducing the geometric setup and highlighting some fundamental differences between these two quantities, I will focus on some structural properties of the set of coexact 1-form spectral gaps of hyperbolic rational homology spheres. In particular, I will discuss a construction that allows to determine somewhat explicitly some interesting accumulation points of the set of such spectral gaps. This is joint work with M. Lipnowski.

March 26th
Speaker: Varinderjit Mann (Cornell University)
Title: Slice enriched categories, microcosm principle, and slice model structures

Abstract: The goal of this talk is to highlight an example of a very important phenomenon known as the microcosm principle, coined by John Baez and James Dolan. This particular example involves discussing pseudomonoid objects and pseudomodule objects. Furthermore, the discussion is interesting in its own right and allows one to define general sliced enriched categories. We will begin by reviewing what it means to categorify general algebraic structures such as monoids. Then, we will take this further to the general context of horizontal categorification and vertical categorification with examples. Our primary consideration of this will be that of a pseudomonoid object and pseudomodule object in a monoidal 2-category. Finally, we discuss the slicing of pseudomonoid objects and pseudomodule objects in the monoidal 2-category Cat, and describe how they respect the consideration of model structures.

April 2nd

(Spring break - no seminar)

April 9th
Speaker: Sanjana Agarwal (Indiana University, Bloomington)
Title: Dennis trace for combinatorial K-theories

Abstract: Classically, algebraic K-theory captures various invariants associated to a ring R used widely in algebraic and arithmetic geometry and number theory. To compute these invariants, one of the most successful tools have been trace methods. The trace method machinery builds off a map called the Dennis trace map from the algebraic K-theory of R to the Hochschild homology of R. In recent years, new analogues of algebraic K-theory have been introduced (first by Zakharevich) in `combinatorial' categories motivated by generalized Hilbert's third problem. In this talk, we present initial attempts to generalize the theory of trace methods to such combinatorial K-theories. This is joint work with Ramyak Bilas.

April 16th
Speaker: Josefien Kuijper (University of Toronto)
Title: All K-theory is squares K-theory: constructing a derived Euler characteristic

Abstract: Combinatorial (or “cut-and-paste”) K-theory is a modern approach to the study of the classical polytope scissors congruence groups, inspired by algebraic K-theory of Waldhausen categories, and can be applied to other geometric settings as well, such as the categories of varieties and semi-algebraic sets. We present the K-theory of squares category as a framework that unifies Waldhausen K-theory as well as many instances of combinatorial K-theory. As an application, we lift the Euler characteristic for definable sets in an o-minimal structure to a map of K-theory spectra.

April 23th
Speaker: Changjie Chen (CRM Montreal)
Title: TBA

Abstract:

April 30th
Speaker: Urshita Pal (University of Michigan)
Title: TBA

Abstract:

Combinatorics Seminar - [SPRING 2026]

2026-04-09T01:24:13-04:00

Combinatorics Seminar

Directions to the department.

Organizers: Laura Anderson, Michael Dobbins, Quaid Iqbal, Tan Tran, and Thomas Zaslavsky.

SPRING 2026

Anticipated speakers: Stefan Viola (Binghamton), Ernesto Estrada (Campus Universitat Illes Balears), Alireza Salahshoori (Binghamton)

Tuesday, 1/20
Organizational meeting
Time: 1:30-2:00 or at most 2:30
Location: WH 100E

Tuesday, 1/27
Speaker: Michael Dobbins (Binghamton)
Title: Transversal Ratios of Convex Polytopes
Time: 1:30-2:30
Location: WH 100E

The transversal ratio of a polytope P is the minimum proportion of vertices of P required to intersect each facet of P. The weak chromatic number of P is the minimum number of colors required to color the vertices of P so that no facet is monochromatic. I will construct an infinite family of d-polytopes for each d at least 5 whose transversal ratio approaches 1 as the number of vertices grows. In particular, this implies that the weak chromatic number for d-polytopes is unbounded for each d at least 5. The previous best known lower bounds on the supremum of the transversal ratio were 1/2 for d=5, 6/11 for d=6, and 2/5 for higher d odd by Novik and Zheng, and 1/2 for higher d even by Holmsen, Pach, and Tverberg.

This is joint work with Seunghun Lee.

Tuesday, 2/3
Speakers: Pratik Misra, Jake Zukaitis, Ali Salahshoori, Quaid Iqbal (Binghamton)
Title: Speed Speaking
Time: 1:30-2:30
Location: WH 100E

Our people will talk about whatever they are thinking about (mathematically).

Tuesday, 2/10
Speaker: Tan Tran (Binghamton)
Title: Signatures in Type A Root Systems and Eulerian Numbers
Time: 1:30-2:30
Location: WH 100E

Let $\Phi$ be a type A root system of rank $n$. In this talk, I introduce a combinatorial invariant, called a “signature”, associated with subsets of $n+1$ positive roots. Given such a subset, its signature records the numbers of positive and negative values among certain determinants obtained by deleting one root at a time. The main construction can be formulated using only elementary linear algebra.

We show that the number of subsets with a given signature can be expressed explicitly in terms of the classical Eulerian numbers. While connections between root systems and Eulerian numbers are well known, this result gives another interpretation of Eulerian numbers in terms of determinant data arising in type A.

These signatures are motivated by questions coming from hyperplane arrangements associated with $\Phi$, including the Shi, Catalan, Linial, and Ish arrangements. As applications, we compute the minimum period of the characteristic quasi-polynomial of these arrangements and determine the values of both the classical and arithmetic Tutte polynomials at (1,1).

This is joint work with M. Cuntz (Hannover), H.M. Tran (Hanoi), and S. Tsujie (Hokkaido).

Tuesday, 2/17
Speaker: M. Seminaire takes a holiday

Tuesday, 2/24
Speaker: Xiyong Yan (Binghamton)
Title: Signs of Hamiltonian Circles in Simple Plane Signed Graphs
Time: 1:30-2:30
Location: WH 100E

I study the signs of Hamiltonian circles in simple plane signed graphs by expressing the sign of a Hamiltonian circle as the product of the signs of the faces it encloses. This leads to the notion of co-Hamiltonian sequences and to a criterion for the existence of Hamiltonian circles of opposite sign, based on sequences whose enclosed face-products have opposite signs. Signed rectangular grid graphs form a natural subclass. For $m \times n$ grids with $m$ even and $m,n>3$, I prove the following theorem. All Hamiltonian circles have the same sign if and only if all non-corner boxes have the same sign. Motivated by the grid structure, I also develop local structural theorems, including ladder-type and hexagon configurations, that guarantee the existence of both positive and negative Hamiltonian circles without explicitly constructing full co-Hamiltonian sequences.

Tuesday, 3/3
Speaker: Patrick Solé (CNRS, Marseille)
Title: Perfect Codes in Weakly Metric Association Schemes
Time: 1:30-2:30
Location: WH 100E and Zoom https://binghamton.zoom.us/j/94050971055?pwd=nmry51RPN2IUT4N2FfK9UxSljuJtqX.1

The Lloyd Theorem of (Solé, 1989) is combined with the Schwartz–Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {polynomial} weakly metric association schemes.

A connection between this notion and the recent theory of multivariate P-polynomial schemes of (Bannai et al., 2025) and of $m$-distance regular graphs (Bernard et al., 2025) is pointed out.

(This is joint work with Minjia Shi and Jing Wang.)

Tuesday, 3/10
Speaker: Aida Abiad (Amsterdam)
Title: On weight-equitable partitions and their application to graph theory
Time: 1:30-2:30
Location: WH 100E and Zoom https://binghamton.zoom.us/j/94050971055?pwd=nmry51RPN2IUT4N2FfK9UxSljuJtqX.1

Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several classic eigenvalue bounds. Weight-equitable partitions assign to each vertex a weight that equals the corresponding entry of the Perron eigenvector. In this talk, I will present several applications of such partitions to graph theory problems.

Tuesday, 3/17
Speaker: Thomas Galvin (Binghamton)
Title: Edge Ideals & Graph Matching
Time: 1:30-2:30
Location: WH 100E

For a simple graph G, the edge ideal I(G) over a field K is a monomial ideal generated by the edges of G. The associated primes of powers of such ideals form an ascending chain. In this talk, I prove a key ingredient of this result, concerning maximum matchings of graphs derived from G. The talk is based on a paper by Martínez-Bernal, Morey, and Villarreal (2012).

Tuesday, 3/24
Speaker: Thomas Martinez (UCLA and Cornell)
Title: The Combinatorics of Affine Deodhar Diagrams
Time: 1:30-2:30
Location: WH 100E

Deodhar diagrams (also known as Go diagrams) have been studied due to their relation to the totally non-negative Grassmannian and positroid varieties. In this talk, we introduce affine Deodhar diagrams and study combinatorial moves of these diagrams. We provide geometric analogues of our combinatorial moves, discussing the relationship to (affine patches of) positroid varieties and their cluster structure.

Tuesday, 4/7
No seminar; it's “Monday”.

Tuesday, 4/14
Speaker: Alexander N. Wilson (York and Binghamton)
Title: Dinv-Zero Fubini Rankings are Pieces of Cake
Time: 1:30-2:30
Location: WH 100E

I will begin with some background on the significance of parking functions in representation theory and some statistics on parking functions arising in this context. I will then focus on a recent project that restricts to a subset of parking functions called Fubini rankings and discuss a surprising result relating enumeration in this subset to the number of regions in a simple hyperplane arrangement.

Tuesday, 4/21
Speaker: Xiyong Yan (Binghamton)
Title: Candidacy exam
Time: 1:30-2:30
Location: WH 100E

This is Mr. Yan's candidacy exam. The examining committee consists of Laura Anderson, Michael Dobbins, and Thomas Zaslavsky (chair). The talks are open to all.

Tuesday, 4/28
Speaker: Alireza Salahshoori (Binghamton)
Title: (Candidacy exam)
Time: 1:30-2:30
Location: WH 100E

This is Mr. Salahshoori's candidacy exam. The examining committee consists of Laura Anderson, Michael Dobbins, and Thomas Zaslavsky (chair). The talks are open to all.

Tuesday, 5/5
Speaker: Danika Van Niel (Binghamton)
Title:
Time: 1:30-2:30
Location: WH 100E

Thursday, 5/7 (Colloquium)
Speaker: Isabella Novik (Washington)
Title:
Time:
Location: WH 100E

FALL 2025

Special Summer Talk
Tuesday, 8/12
Speaker: Seunghun Lee (Institute for Basic Science)
Title: Rainbow Theorems in Oriented Matroids
Time: 1:15-2:15
Location: WH 100E

I will present some new rainbow theorems in oriented matroids. In particular, the rainbow tope conjecture, which was originally motivated from a graph-theoretic conjecture by Yu Yokoi, will be presented with some partial results.

This is joint work with Minho Cho and Frederic Meunier.

Tuesday, 8/19
Organizational meeting
Time: 1:30-2:30
Location: WH 100E

Tuesday, 8/26
Speaker: Quaid Iqbal, Michael Dobbins, Marwa Mosallam, Alireza Salahshoori (Binghamton)
Title: What We Do: An Introduction
Time: 1:30-2:30
Location: WH 100E
We each spend 10 minutes telling you what we do in math. Questions are welcome.

Thursday, 9/4 (Note special day and special room.)
Speaker: M. Tariq Rahim (Abbottabad University of Science & Technology)
Title: Some Extremal Properties of a Topological Index Based on Degree and Distance in Graphs
Time: 1:30-2:30
Location: WH 309

Topological indices of graphs, which are computed from numbers associated to graph structure, have been an active area of research due to their wide range of applications in chemistry and their mathematical depth. In recent years, several new topological indices have been introduced, and various properties of these indices have been thoroughly investigated. However, much of the research has focused on indices utilizing the degrees of vertices of the graph.

I introduce a topological index that incorporates both the distances and the degrees in a graph. We compute the minimum values of this index and characterize the unique extremal graph attaining the minimum value.

Tuesday, 9/9
Speaker: Xiyong Yan (Binghamton)
Title: Double Signs of Hamiltonian Circles in Doubly Signed Complete Graphs
Time: 1:30-2:30
Location: WH 100E

We study Hamiltonian circles in the doubly signed complete graph $\Sigma_n = (K_n, \sigma, \mathbb{F}_2^2)$. A circle’s double sign is defined as the sum of its edge labels. I establish conditions under which Hamiltonian circles realize all four possible double signs and prove that this occurs when the set of triangle double signs contains at least three distinct values. The proof is based on an analysis of triangle bases of the binary cycle space, structural properties of $K_4$ subgraphs, and explicit Hamiltonian constructions.

Tuesday, 9/16
Speaker: Jaiung Jun (New Paltz)
Title: Tropical Subrepresentions and Matroids
Time: 1:30-2:30
Location: WH 100E
In their recent paper, Giansiracusa and Manaker introduced a notion of tropical subrepresentations of linear representations by considering group actions on tropical linear spaces. In particular, this framework naturally brings matroids into the picture. I will present several elementary results on tropical subrepresentations.

This is joint work with Kalina Mincheva and Jeffrey Tolliver.

Tuesday, 9/23
No seminar (Rosh Hashonah holiday)

Tuesday, 9/30
Speaker: Alireza Salahshoori (Binghamton)
Title: Circle Gain Systems
Time: 1:30-2:30
Location: WH 100E
A “circle gain system” is a labeling of all oriented rooted circles in the graph $\Gamma$ with elements from the group $\mathcal{G}$. In this talk, we will see the necessary and sufficient conditions for a circle gain system to be realizable by a gain graph.

Tuesday, 10/7
There will be no seminar today.
Time: 1:30-2:30
Location: WH 100E

Tuesday, 10/14 (jointly with the Geometry/Topology Seminar)
Speaker: Lee Kennard (Syracuse)
Title: Regular Matroids and Torus Representations
Time: 1:30-2:30
Location: WH 100E
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.

Tuesday, 10/21
No seminar today.
Location: WH 100E

Tuesday, 10/28
The seminar takes a holiday today.
Location: WH 100E

Tuesday, 11/4 Cancelled
Speaker: Brendon Rhoades (U.C. San Diego)
Title: Matrix Loci and Shadow Play
Time: 1:30-2:30
Location: WH 100E

Let lis$(w)$ be the length of the longest increasing subsequence of a permutation $w$ in $S_n$. I describe a graded quotient $R_n$ of the polynomial ring over an $n$-by-$n$ matrix of variables whose Hilbert series is the generating function of lis, up to reversal. The Gröbner theory of $R_n$ is governed by the Viennot shadow avatar of the Schensted correspondence. The ring $R_n$ is constructed via the orbit harmonics technique of deformation theory.

I will give some related results and open problems.

Joint with Jasper Liu, Yichen Ma, Jaeseong Oh, and Hai Zhu.

Tuesday, 11/11
Speaker: Tan Tran (Binghamton)
Title: Vine Copulas, MAT-Labeled Graphs, and Single-Peaked Domains: A Three-Way Correspondence
Time: 1:30-2:30
Location: WH 100E

Last year, I discussed an unexpected link between vine copulas—graphical models used in statistics—and MAT-labeled graphs, which arise in algebraic graph theory through the study of free hyperplane arrangements.

This year, I’ll add a third piece to the picture. With H. M. Tran and S. Tsujie, I recently found that these structures are also closely connected to single-peaked domains in voting theory. In particular, MAT-labeled complete graphs, regular vines, and maximal Arrow’s single-peaked domains turn out to be three different manifestations of the same underlying combinatorial framework. This connection brings together ideas from algebraic combinatorics, probabilistic modeling, and social choice. As a consequence of this correspondence, we obtain a complete combinatorial characterization of maximal Arrow’s single-peaked domains, resolving a recent open question in the economics community.

Tuesday, 11/18 (joint with the Arithmetic Seminar)
Speaker: Jaeho Shin (Seoul National University)
Title: Biconvex Polytopes and Tropical Linear Spaces
Time: Special time 4:00-5:00
Location: WH 100E

Tropical geometry is geometry over exponents of algebraic expressions, using the “logarithmized” operations (min,+) or (max,+). In this setting, one can define tropical convexity and the related notion of biconvex polytopes, which are convex both classically and tropically. There is also a tropical analogue of linear spaces, called tropical linear spaces. Sturmfels conjectured that every biconvex polytope arises as a cell of a tropical linear space. In this talk, I will outline a proof of this conjecture.

Tuesday, 11/25
Speaker: Xiyong Yan (Binghamton)
Title: Realization and Classification of Hamiltonian-Circle Multisigns
Time: 1:30-2:30
Location: WH 100E

We investigate the multisigns of Hamiltonian circles in the multisigned complete graph $\Sigma_n := (K_n, \sigma, \mathbb{F}_2^m)$. For a fixed $m$ and sufficiently large $n$, I prove that the set of multisigns of Hamiltonian circles $\{\sigma(H) : H \text{ is a Hamiltonian circle of } \Sigma_n\}$ forms either a subspace, an affine subspace, or the entire space $\mathbb{F}_2^m$, except in certain exceptional cases.

The main tools used are the $C_4$ Necklace Lemma and triangular paths.

Tuesday, 12/2
Another working holiday today.
Location: WH 100E

SPRING 2025

Tuesday, 1/21
Organizational meeting
Time: 1:15-2:00
Location: WH 100E

Tuesday, 1/28
Speaker: Ryan McCulloch (Binghamton)
Title: Incidence Gain Graphs and Generalized Quadrangles
Time: 1:15-2:15
Location: WH 100E

Our setting is that we have an incidence structure (P, B, I) and $\Gamma$ its incidence graph. We also have a gain graph $(\Gamma,\varphi)$ where the gain group acts on some nonempty set $\Lambda$. I provide a general construction that takes such an “incidence gain graph” and produces an incidence structure (P', B', I'). I give conditions for when (P', B', I') is a generalized quadrangle. I then explicitly build generalized quadrangles for any affine plane over a field. To this end, I take the additive group action of the field on itself. I can define a gain function that satisfies all of the conditions needed to guarantee a generalized quadrangle. The construction works for any affine plane over any field, whether finite or infinite.

Tuesday, 2/4
Speaker: Alireza Salahshoori (Binghamton)
Title: Circle Gains in Gain Graphs
Time: 1:15-2:15
Location: WH 100E

A gain graph is a graph in which each oriented edge is assigned a gain, represented by a group element, with the gain inverted when the edge orientation is reversed. The gain of a circle in a gain graph is defined as the product of the gains of its edges, with the result depending on the orientation and the chosen “root” (starting edge) of the circle. A “circle gain system” refers to the collection of all oriented and rooted circle gains. In this talk I establish the necessary and sufficient conditions for a circle gain system to originate from a gain graph, specifically when the underlying group is abelian.

Tuesday, 2/11
Speaker: Xiyong Yan (Binghamton)
Title: Hamiltonian Cycles in Signed and Multisigned Complete Graphs
Time: 1:15-2:15
Location: WH 100E

A signed complete graph contains both positive and negative Hamiltonian cycles if and only if it also contains both positive and negative triangles. Otherwise, all Hamiltonian cycles are negative if and only if all triangles are negative and n is odd, while all Hamiltonian cycles are positive if and only if all triangles are negative and n is even, or all triangles are positive. Extending these results to multisigned complete graphs, I prove that such a graph contains at least two Hamiltonian cycles with different multisigns if and only if it contains at least two triangles with different multisigns.

Tuesday, 2/18
Speaker: Thomas Zaslavsky (Binghamton)
Title: What a Projective Rectangle “Really” Is
Time: 1:15-2:15
Location: WH 100E

Rigoberto Florez and I invented “projective rectangles” in 2005 and only recently found out that all finite ones were beautifully classified in terms of “attenuated spaces” over finite fields in a 1977 paper about partial geometries. I will explain the meaning of these terms and the classification (without the lengthy proof). Questions remain that we have not yet dealt with.

Tuesday, 2/25
Special Event
ADMA Colloquium Lecture by Pavol Hell (Simon Fraser University)
“Signed Graphs, Flows and Homomorphisms”
8:30 a.m. - 9:30 a.m. EST
On Zoom; registration required (see T. Zaslavsky)

ADMA is the Academy of Discrete Mathematics and Applications, in India. The talk will be a survey.

Signed graphs first arose in the theory of social balance, but also in clustering in networks, in root systems, matroids, flows on non-orientable surfaces, and in many other areas. A signed graph is an undirected graph G together with a labeling of its edges by signs + and −, where two signed graphs are considered the same if one can be obtained from the other by a sequence of switchings (flipping all +,− signs at a vertex). The topic of signed graphs has long been championed by T. Zaslavsky, and more recently by R. Naserasr and E. Sopena. In this expository talk I will illustrate how natural the concept of signed graphs is, and how it oﬀers a refined view of many basic graph theory results. This phenomenon will be further underscored by presenting new results on nowhere zero flows and on the complexity of homomorphism problems. Results of M. Devos, K. Nurse, R. Šámal, J. Bok, R. Brewster, T. Feder, N. Jedličková, H. Kim, A. Rafiey, M. Siggers, the speaker, and others will be discussed.

Tuesday, 2/25
Speaker: Tran Tan (Binghamton)
Title: Inductive and Divisional Posets: A Study of Poset Factorability
Time: 1:15-2:15
Location: WH 100E

I introduce and study the notion of inductive posets and their superclass, divisional posets, inspired by the concepts of inductive and divisional freeness for central hyperplane arrangements. A poset is called factorable if its characteristic polynomial has all positive integer roots. Motivated by this, we define inductive and divisional abelian (Lie group) arrangements, with their posets of layers serving as primary examples. Our first main result shows that every divisional poset is factorable. The second result establishes that the class of inductive posets includes strictly supersolvable posets, a class recently introduced by Bibby and Delucchi (2024), which extends the classical result by Jambu and Terao (1984) that every supersolvable hyperplane arrangement is inductively free. Finally, we present an application to toric arrangements, proving that the toric arrangement defined by any ideal of a root system of type A, B, or C, with respect to the root lattice, is inductive.

This work is joint with R. Pagaria (Bologna), M. Pismataro (Bologna), and L. Vecchi (KTH).

Tuesday, 3/4
Speaker: Himanshu Gupta (Regina)
Title: Minimum Number of Distinct Eigenvalues of Johnson and Hamming Graphs
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/94050971055?pwd=nmry51RPN2IUT4N2FfK9UxSljuJtqX.1

This talk focuses on the inverse eigenvalue problem for graphs (IEPG), which seeks to determine the possible spectra of symmetric matrices associated with a given graph $G$. These matrices have off-diagonal non-zero entries corresponding to the edges of $G$, while diagonal entries are unrestricted. A key parameter in IEPG is $q(G)$, the minimum number of distinct eigenvalues among such matrices.

The Johnson and Hamming graphs are well-studied families of graphs with many interesting combinatorial and algebraic properties. We prove that every Johnson graph admits a signed adjacency matrix with exactly two distinct eigenvalues, establishing that its $q$-value is two. Additionally, we explore the behavior of $q(G)$ for Hamming graphs.

This is a joint work with Shaun Fallat, Allen Herman, and Johnna Parenteau.

Thursday, 3/6 (Special day)

Speaker 1 of 2: Norihiro Nakashima (Nagoya Tech)
Title: Characteristic Quasi-Polynomials of the Restriction of Shi Arrangement
Time: 2:50-3:50
Location: WH 329 (Special room)

The characteristic quasi-polynomial counts the number of elements in the complement of a hyperplane arrangement modulo a positive integer. In this talk, we will introduce a method for computing the characteristic quasi-polynomials of arrangements obtained by restricting Shi arrangements of type B and type C to a single hyperplane. We will also determine whether period collapse occurs in the characteristic quasi-polynomial under deletion of a hyperplane in the Shi arrangement.

This talk includes joint work with Higashitani and joint work with Ono.

Speaker 2 of 2: Takuro Abe (Rikkyo)
Title: Graphic Arrangements and Logarithmic Modules
Time: 4:25-5:25
Location: WH 329 (Special room)

By associating a hyperplane to each edge of a graph, there is a one to one correspondence between graphs and subarrangements of the braid arrangement, which is called a graphic arrangement. One of the main problem among graphic arrangements is to ask which property of them are characterized by combinatorics of graphs. One of the most successful results is due to Stanley, characterizing the freeness of graphic arrangements in terms of chordal graphs.

In this talk, we give the next step of Stanley's classical result in terms of projective dimensions. Namely, we show that a logarithmic derivation module is of projective dimension at most one if and only if the corresponding graph is weakly chordal. We also give several problems related to it.

This is a joint work with Lukas Kuehne, Paul Muecksch and Leonie Muehlherr.

Dinner will be held after the talks; anyone who wishes to join is welcome. See the organizers after the talks are concluded, but please register your interest before noon with Tan Tran (tnhattran@binghamton.edu) so he can ensure we get a reservation.

Tuesday, 3/18
Speaker: Michael Dobbins (Binghamton)
Title: Neighborhoods of Rank 3 Topological Representations
Time: 1:15-2:15
Location: WH 100E

MacPhersonians are combinatorial analogs of Grassmannians that are defined by oriented matroids. Pseudolinear Grassmannians are spaces of topological representations of oriented matroids, and these are each homotopy equivalent to the corresponding Grassmannian in rank 3. I will present a good cover of the rank 3 pseudolinear Grassmannians with nerve complex isomorphic to the order complex of the corresponding MacPhersonian.

Tuesday, 3/25
Speaker: Vishal Gupta (Delaware)
Title: Minimum Spectral Radius in a Given Class of Graphs
Time: 1:15-2:15
Location: WH 100E

In 1986, Brualdi and Solheid posed the question of determining the maximum and minimum spectral radius of a graph within a given class of simple graphs. Since then, this problem has been extensively studied for various graph classes. I will discuss two such classes: simple connected graphs with a given order and size, and simple connected graphs with a given order and dissociation number.

This presentation is based on joint work with Sebastian Cioaba, Dheer Noal Desai, and Celso Marques.

Tuesday, 4/1
Cancelled due to travel problems
Speaker: Ulysses Alvarez (Alabama)
Title: Clusters, Cells and Triangulations
Time: 1:15-2:15
Location: WH 100E

Cluster algebras are commutative rings with a set of distinguished generators called cluster variables. In 2001, Fomin and Zelevinsky showed that every cluster variable in any cluster algebra is a Laurent polynomial with integer coefficients. In 2006, it was shown by Caldero and Zelevinsky that the coefficients of cluster variables are precisely the Euler characteristic of an associated quiver Grassmannian. In order to more easily compute these coefficients, we are interested in finding cell decompositions for these topological spaces. I will discuss the coefficients for the special class of quivers obtained by triangulations of bordered surfaces and how we can use what we know in this setting to potentially find cell decompositions for more general quivers.

Tuesday, 4/8
Speaker: Mutasim Mim (City University of New York)
Title: Clique Complexes of Strongly Regular Graphs and Their Eigenvalues
Time: 1:15-2:15
Location: WH 100E

It is known that non-isomorphic strongly regular graphs with the same parameters must be cospectral (have the same eigenvalues). In this paper, we investigate whether the spectra of higher order Laplacians associated with these graphs can distinguish them. In this direction, we study the clique complexes of strongly regular graphs, and determine the spectra of the triangle complexes of several families of strongly regular graphs including Hamming graphs and Triangular graphs. In many cases, the spectrum of the triangle complex distinguishes between strongly regular graphs with the same parameters, but we find some examples where that is not the case.

This is joint work with Sebastian Cioaba, Krystal Guo, and Chunxu Ji.

Tuesday, 4/15
Speaker: Marie Kramer (Syracuse)
Title: Graph Embeddings & Torus Obstructions
Time: 1:15-2:15
Location: WH 100E

While obstructions to embedding graphs into the plane and the real projective plane are well understood, there is no known complete list for other surfaces such as the torus or the Klein bottle. The embeddability of planar obstructions into other surfaces has been studied: Mohar and Gagarin, Kocay, and Neilson classified such embeddings into the real projective plane and the torus, respectively.

I will discuss a similar result classifying the embeddings of the six cubic projective plane obstructions into the torus. I will show how this result helped us verify Chambers’s computer-assisted results regarding cubic torus obstructions with small first Betti number.

Lastly, I will highlight a connection between the existence of embeddings of graphs into certain surfaces and geometric data of special torus representations found by Kennard, Wiemeler, and Wilking.

Tuesday, 4/29 (joint with the Geometry/Topology Seminar)
Speaker: Leo Jiang (Toronto)
Title: Topology of Real Matroid Schubert Varieties
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/94050971055 Passcode 303207

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.

Tuesday, 5/6
Speaker: Jake Zukaitis (Binghamton)
Title 1: Automorphisms That Fix Elements of a Switching Class of a Mixed Graph
Title 2: 2-Structures, Switching Classes, and Primitivity
Time: 1:15-2:15 and 2:50-3:50
Location: WH 100E and WH 309

Talk 1: Mixed graphs are graphs where some edges have a direction and some do not. Switching means partitioning the vertex set into two parts, and changing the edges between the two parts, as follows: for directed edges, changing direction, and for undirected edges, changing existence. Switching is an equivalence relation on graphs. A generalization is to only add edges in the latter case that are in some “universal” graph.

Automorphisms of this “universal” graph can be applied to mixed graphs; the question of fixed elements comes up. Sometimes an automorphism can fix an equivalence class of switching. I will show necessary and sufficient conditions for an automorphism to fix a specific element of every equivalence class it fixes.

This is based on the paper “Switching Classes of Directed Graphs and $H$-Equivalent Matrices” by Ying Cheng.

Talk 2: A 2-Structure is an equivalence relation on a finite set $V^2$ without the diagonal. Call this set $V_2$. A Clan of a 2-Structure is a subset of $V$, call it $X$, with the property that for every pair $x,y \in X$ and every $v$ not in $X$, then $(x,v)$ and $(y,v)$ are in the same equivalence class. An $A$-labeled 2-Structure, where $A$ is an abelian group, is a 2-Structure where each equivalence class is assigned a group element. A useful way to view an $A$-labeled 2-Structure is with the function $g: V_2 \to A$ sending each ordered pair to the element assigned to the equivalence class containing that element. A Clan of $g$ is then a set $X$ where for every pair $x,y \in X$ and every $v$ not in $X$, $g(x,v) = g(y,v)$.

Switching classes can be defined in a way analogous to the previous talk. A Primitive Switching Class is a switching class of a 2-structure with the minimum number of clans. The goal of this talk is to introduce a Primitivity Graph and show how, under one additional assumption, this graph is a cycle with at most one isolated vertex.

This will be Mr. Zukaitis's candidacy exam. The examining committee will be Thomas Zaslavsky (chair), Laura Anderson, and Michael Dobbins. The lectures are open to all and all are welcome.

FALL 2024

Tuesday, 8/20
Organizational meeting
Time: 1:15-2:00
Location: WH 100E
Zoom link: https://binghamton.zoom.us/j/95157730373

Tuesday, 8/27
Speakers: Laura Anderson, Quaid Iqbal, Ryan McCulloch, Tan Nhat Tran, Jake Zukaitis
Title: Like speed dating, but combinatorics
Time: 1:15-2:15
Location: WH 100E

A few of us will say a few words about what we're thinking about these days.

Tuesday, 9/3
Speakers: Michael Dobbins, Tara Koskulitz, Ali Salahshoori, Stefan Viola, Thomas Zaslavsky
Title: Like speed dating, but combinatorics
Time: 1:15-2:15
Location: WH 100E

A few more of us will say a few words about what we're thinking about these days.

Tuesday, 9/10
Speaker: Tan Nhat Tran (Binghamton)
Title: Algebraic Combinatorics Meets Probability Theory: Vines and MAT-Labeled Graphs, Part I
Time: 1:15-2:15
Location: WH 100E

The present talk discusses a connection between two concepts arising from different fields of mathematics. The first concept, called a vine, is a graphical model for dependent random variables. This first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of the logarithmic derivation module of hyperplane arrangements. This was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards developed systematically by Cuntz-Muecksch (2020).

In the particular case of graphic arrangements, Tsujie and I recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. I show that, interestingly, there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. In particular, we obtain an equivalence between the categories of regular vines and MAT-labeled complete graphs.

This is joint work with H.M. Tran (Singapore) and S. Tsujie (Hokkaido).

Tuesday, 9/17
Speaker: Tan Nhat Tran (Binghamton)
Title: Algebraic Combinatorics Meets Probability Theory: Vines and MAT-Labeled Graphs, Part II
Time: 1:15-2:15
Location: WH 100E

Tuesday, 9/24
Speaker: Michael Dobbins (Binghamton)
Title: Asymptotically Counting Nerves and VC-Dimension
Time: 1:15-2:15
Location: WH 100E

How many graphs can be represented as the intersection graph of unit intervals, or more generally, how many simplicial complexes can be represented as the nerve of a family of unit balls?

I will present rough asymptotic bounds on the number of simplicial complexes that can be represented as the nerve of a family of convex sets belonging to certain classes. In particular, we will look at classes consisting of translates of a semialgebraic convex set, such as a unit ball. In this case, the number of possible nerves is much lower than for convex sets in general. The argument will use VC-dimension, but just bounding VC-dimension is not enough. We will see a class of convex sets of bounded VC-dimension where the number of possible nerves is close to that for convex sets in general.

This work is with Minho Cho, Boris Bukh, Amzi Jeffs, Jinha Kim, Minki Kim, and Shiyi Ma.

Tuesday, 10/1
No meeting: Rosh Hashonah holiday

Tuesday, 10/8
No meeting: Friday classes

Tuesday, 10/15
Speaker: Nicholas Proudfoot (Oregon and Oxford)
Title: Kazhdan–Lusztig Polynomials of Complete Graphs
Time: 1:15-2:15
Location: WH 100E and Zoom https://uoregon.zoom.us/j/5484160691

I will introduce the notion of the Kazhdan–Lusztig polynomial of a graph (more generally a matroid, but I’ll stick with graphs for concreteness). I’ll then present two families of examples, namely cycles and complete graphs. The solution for complete graphs, due to Ferroni and Larson, involves the enumeration of series-parallel graphs. I’ll end with an open problem about how to incorporate symmetries of the complete graph into the story.

Tuesday, 10/22
Speaker: Caitlin Lienkaemper (Boston University)
Title: Using oriented matroids to find low rank structure in presence of nonlinearity
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/94050971055?pwd=nmry51RPN2IUT4N2FfK9UxSljuJtqX.1

Estimating the linear dimensionality of a data set in the presence of noise is a common problem. However, data may also be corrupted by monotone nonlinear distortion that preserves the ordering of matrix entries but causes linear methods for estimating rank to fail. In light of this, we consider the problem of computing underlying rank, which is the lowest rank consistent with the ordering of matrix entries, and monotone rank, which is the lowest rank consistent with the ordering within columns. We show that each matrix of monotone rank $d$ corresponds to a point arrangement and a hyperplane arrangement in $\mathbb R^d$, and that the ordering within columns of the matrix can be used to recover information about these arrangements. Using Radon's theorem and the related concept of the VC dimension, we can obtain lower bounds on the monotone rank of a matrix. However, we also show that the monotone rank of a matrix can exceed these bounds. In order to obtain better bounds on monotone rank, we develop the connection between monotone rank estimation and oriented matroid theory. Using this connection, we show that monotone rank is difficult to compute: the problem of deciding whether a matrix has monotone rank two is already NP-hard. However, we introduce an “oriented matroid completion” problem as a combinatorial relaxation of the monotone rank problem and show that checking whether a set of sign vectors has matroid completion rank two is easy.

Tuesday, 10/29
Speaker: Sebastian Cioaba (Delaware)
Title: Spectral Moore Theorems for Graphs and Hypergraphs
Time: 1:15-2:15
Location: WH 100E

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph, which is the difference between its valency and second eigenvalue, is widely seen as an algebraic measure of connectivity and plays a key role in the theory of expander and Ramanujan graphs. In this talk, I will give an overview of recent work studying the maximum order of a regular graph (bipartite graph or hypergraph) of a given valency whose second largest eigenvalue is at most a given value. This problem can be seen as a spectral Moore problem and has close connections to Alon–Boppana theorems for graphs and hypergraphs and with the usual Moore or degree-diameter problem.

Tuesday, 11/5
Speaker: Jake Zukaitis (Binghamton)
Title: A Nonstandard Notion of Connection for Signed Graphs
Time: 1:15-2:15
Location: WH 100E

Connection usually means the existence of paths between vertices of a graph. In Signed Graph Theory, circles that have positive sign (called balanced circles) play an important role. I will define a different notion of connection, called balanced-circle-connection, which, as the name implies, will incorporate these balanced circles. Afterward, I will show alternate characterizations of this property.

Tuesday, 11/12
Speaker: Gang Zhou (Binghamton)
Title: On the Continuity of Magnetization of the 3-Dimensional Square-Lattice $XY$ Model
Time: 1:15-2:15
Location: WH 100E

I will present a preliminary result with Juerg Froehlich. Under certain assumptions on the directed graphs generated by the random path representation of the $XY$ model, we invented a switching lemma and proved the continuity of magnetization of the 3-dimensional square-lattice $XY$ model at the critical temperature.

I will present the combinatorics about the graphs; the reason we believe our assumption holds. Any new insights are appreciated.

For interested parties, I will discuss the importance of this type of problem.

Tuesday, 11/19
Speaker: Tan Tran (Binghamton)
Title: Shi and Ish
Time: 1:15-2:15
Location: WH 100E

The Shi arrangement of hyperplanes arose in algebraic geometry and has become a significant inspiration in combinatorial geometry. The Ish arrangement is a kind of inversion of the Shi arrangement that has strangely similar properties. With a main emphasis on freeness and supersolvability, I will first survey what is known in type A and then introduce an extension to type B.

This is based on joint work with T. Abe (Tokyo) and S. Tsujie (Hokkaido).

Tuesday, 11/26
No meeting: Friday classes

Tuesday, 12/3
Speaker: Benjamin Braun (Kentucky)
Title: Just Relax and Go with the Flow (Polytopes)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/s/94050971055 Passcode 303207

Flow polytopes are beautiful geometric objects that encode the different ways material can move through a transportation network. Flow polytopes for certain types of transportation networks have particularly nice volumes and geometric structure. We will take a tour through the world of flow polytopes, starting with the basics for those who have never heard of them and ending by highlighting several recent results by faculty and students at the University of Kentucky.

SPRING 2024

Tuesday, 1/16
No meeting; it's too soon.

Tuesday, 1/23
Organizational meeting
Time: 1:15-2:00
Location: WH 100E

Tuesday, 1/30
No meeting this week.

Tuesday, 2/6
No meeting this week.

Tuesday, 2/13
Speaker: Michael Dobbins (Binghamton)
Title: Colorful Intersections and Tverberg Partitions
Time: 1:15-2:15
Location: WH 100E

Consider 6 convex bodies in 3-space, 3 red and 3 blue, such that each red-blue pair intersects. Then, either there must be a line through all 3 red bodies or through all 3 blue bodies. With this observation as a starting example, we show that if m families of k+r convex bodies each in d-space have the colorful intersection property, and if d<(r+1)m/(k-1) and k is a prime power, then one of the families is intersected by an affine r-flat. Moreover, we prove an interpolation between the colorful Helly theorem and Tverberg’s theorem. As part of the proof we use discrete Morse theory to analyse the connectivity of a certain simplicial complex of partitions.

This is joint work with Andreas Holmsen and Dohyeon Lee.

Tuesday, 2/20
Speaker: Alexander Vidinas (Cornell)
Title: On the Alexander Polynomial of Special Alternating Links
Time: 1:15-2:15
Location: WH 100E

The Alexander polynomial (1928) is the first polynomial invariant of links devised to help distinguish links up to isotopy. In recent work of Elena Hafner, Károla Mészáros, and the speaker, Fox's conjecture (1962) – stating that the absolute values of the coefficients of the Alexander polynomial for any alternating link are unimodal – was settled for special alternating links. The present talk outlines a study of the special combinatorial and discrete geometric properties that Alexander polynomials of special alternating links possess along with a generalization to all Eulerian graphs, introduced by Murasugi and Stoimenow (2003). We prove that the Murasugi and Stoimenow generalized Alexander polynomials can be expressed in terms of volumes of root polytopes of unimodular matrices, building on the beautiful works of Li and Postnikov (2013) and Tóthmérész (2022). We conjecture a generalization of Fox's conjecture to the Eulerian graph setting. We also bijectively relate two longstanding combinatorial models for the Alexander polynomials of special alternating links: Crowell's state model (1959) and Kauffman's state model (1982, 2006).

Tuesday, 2/27
Speaker: Laura Anderson (Binghamton)
Title: Triangulations of Oriented Matroids
Time: 1:15-2:15
Location: WH 100E

A triangulation of an oriented matroid is a combinatorial analog to a geometric triangulation of the convex hull of a set of points in affine space. Even finding a good definition of oriented matroid triangulation is surprisingly tricky, and the most fundamental conjecture – that such a triangulation should be a topological ball – has been open for many years.

This talk is a historical survey, as well as an appeal for a new generation to take up the quest.

Tuesday, 3/5
Spring break!

Tuesday, 3/12
Speaker: Yichen Ma (Cornell)
Title:: Invariants of Partial Orders and Convex Geometries via Characters on Hopf Monoids
Time: 1:15-2:15
Location: WH 100E

We consider a Hopf monoid of partial orders and another of convex geometries, and investigate combinatorial invariants constructed from characters on them. Each invariant comes in a pair consisting of a polynomial and a (more general) quasisymmetric function.

For partial orders we obtain the order polynomial of Stanley and the enriched order polynomial of Stembridge. For convex geometries we obtain polynomials first introduced by Edelman-Jamison and Billera-Hsiao-Provan. We obtain reciprocity results satisfied by these polynomials from the perspective of characters in a unified manner.

We also describe the coefficients of the quasisymmetric invariants as enumerating faces on certain simplicial complexes. These include the barycentric subdivision of the CW-sphere of a convex geometry introduced by Billera, Hsiao and Provan.

Tuesday, 3/19
Speaker: Thomas Zaslavsky (Binghamton)
Title: Correlation Clustering: Signed Graphs, Algorithms, and a Best Case
Time: 1:15-2:15
Location: WH 100E

A signed graph has its edges labelled positive and negative; we regard an edge as agreement ($+$) and disagreement ($-$) between its endpoints. A clustering is a partition of the vertex set into subsets, called “clusters”. Correlation clustering, introduced by Bansal et al., wants all edges within a cluster to represent agreement ($+$) and all edges between clusters to represent disagreement ($-$), but that is rarely possible, so it seeks to minimize the number of “bad” edges: positive edges between clusters and negative edges within clusters; this minimum is the “clusterability index” $Q$ and its realization is an “optimal clustering”. Finding $Q$ or an optimal clustering is NP-hard, but there is a simple lower bound on $Q$ which is attained under certain conditions. The signed graphs that meet those conditions, and their optimal clusterings, can be described precisely.

This work is joint with Leila Parsaei-Majd and Michael Gottstein. The talk can be regarded as an introduction to signed graphs.

Tuesday, 3/26
Speaker: Peter Maceli (Ithaca)
Title: Structure of Self-Complementary Graphs
Time: 1:15-2:15
Location: WH 100E

A graph is called self-complementary if it and its complement are isomorphic. The class of self-complementary graphs is structurally and algorithmically very rich, yet little is known about decomposing or explicitly constructing such graphs. I will discuss a structural conjecture of Trotignon, as well as a number of general techniques for constructing self-complementary graphs.

Tuesday, 4/2
There will be no seminar today as we prepare for the solar eclipse. Remember not to look at the 97%-eclipsed sun without adequately dark glasses.

Tuesday, 4/9
Speaker: Quaid Iqbal (Binghamton)
Title: Description of Distance-Regular Graphs with Fixed Parameters
Time: 1:15-2:15
Location: WH 100E

A graph $\Gamma$ is distance regular if, for any two vertices $v$ and $w$ at distance $d$, the number of vertices at distance $j$ from $v$ and distance $k$ from $w$ depends only on $d, j$, and $k.$ Distance-regular graphs have very nice eigenvalue properties (e.g., they have $d+1$ distinct eigenvalues) and are a lively topic in spectral graph theory. Distance-regular graphs with diameter $2$, which are called strongly regular, have been important in the classification of finite groups. Given a graph $\Gamma$, the distance-$2$ graph $\Gamma_2$ is the graph on the same vertices, in which vertices are adjacent if they have distance $2$. I consider the distance-regular graphs $\Gamma$ whose distance-$2$ graphs $\Gamma_2$ are strongly regular. If $\Gamma$ is bipartite, then its distance-$2$ graph is not connected. So, I am interested in the class of non-bipartite distance-regular graphs. I explain that it can be described with a fixed parameter, that is by eigenvalue or by intersection number. First, I will show that the distance-$2$ graph of a non-bipartite distance-regular graph with diameter $D=3,4$ and eigenvalue $a_ {2}-c_ {3} $ is strongly regular, and then I will give several kinds of descriptions of non-bipartite distance-regular graphs with diameter $D=3,4$ and eigenvalue $a_ {2}-c_ {3}$ under various conditions, for example when the $\tilde{c} =p$ (prime) (where $\tilde{c}$ is the number of common neighbors between any two non-adjacent vertices).

Tuesday, 4/16
Speaker: Michael Gottstein (Binghamton)
Title: Partitions and Gain Graphs
Time and Location: 1:15-2:15 in WH 100E and 3:00-4:00 in WH 309

The Rhodes semilattice of a group is a fundamental tool used in the complexity theory of finite semigroups. I reinterpret the Rhodes semilattice into the language of gain graphs. This reinterpretation naturally suggests several lattice extensions of the Rhodes semilattice. One of these lattices can be seen as a vast generalization of finite groupoids. The objective of this defense is to demonstrate and support this point of view.

This is Mr. Gottstein's Ph.D. thesis defense. The examining committee consists of Laura Anderson, Michael Dobbins, Leslie Lander (outside examiner), and Thomas Zaslavsky (advisor and chair). The presentation is open to the public and all are welcome to attend.

Tuesday, 4/30
Speaker: Amena Assem (University of Toronto Mississauga)
Title: Progress Towards the Orientation Conjecture of Nash-Williams for Infinite Graphs
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/3475600721

Nash-Williams proved in 1960 that an edge connectivity of 2k is sufficient for a finite graph to admit a k-arc-connected orientation and conjectured that the same holds for infinite graphs. I show that the conjecture is true for an important class of infinite graphs.

This is joint work with Max Pitz and Marcel Koloschin from the University of Hamburg.

Tuesday, 5/7
Speaker: Fiona Young (Cornell)
Title: The essential bound of a $k$-polymatroid and applications to excluded minor problems
Time: 1:15-2:15
Location: WH 100E

The singleton and doubleton minors of a polymatroid encode a surprising amount of information about its structural complexity. Starting with a $k$-polymatroid $\rho$, we subtract from it as many maximally-separated matroids as possible. Let the result be an $m$-polymatroid; this gives rise to a notion of boundedness for $\rho$. When $k$ is sufficiently large, the bounds on the singleton and doubleton minors of $\rho$ completely determine the bound on $\rho$. Much of this is motivated and guided by the polytopal perspective of polymatroids. Our results provide an organized framework for thinking about polymatroid excluded minor problems.

FALL 2023

Tuesday, 8/29
Organizational meeting
Time: 1:15-2:00
Location: WH 100E

Tuesday, 9/12
Title: Like Speed Dating, But Combinatorics
Speakers: Michael Gottstein, Tara Koskulitz, and Stefan Viola (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Michael Gottstein, Tara Koskulitz, and Stefan Viola will briefly describe what they're working on these days.

Tuesday, 9/19
Title: The “Monotonic Linear model” Via Oriented Matroids
Speaker: Laura Anderson (Binghamton)
Time: 1:15-2:15
Location: WH 100E

This talk concerns models of the form $\mathbf y=F(\mathbf A\mathbf x)$ where $\mathbf y$ can be measured by an experimenter, $\mathbf A$ is a known matrix, $\mathbf x$ can be manipulated by the experimenter but not measured, and $F=(f,f,\ldots,f)^\top$ is an unknown componentwise increasing function.

Part 1 of the talk will explain how such models arise in psychology and why they are hard to test. Part 2 will discuss how testing such a model amounts to testing “conformity to an oriented matroid”. Part 3 will describe a new statistical procedure to carry out such a test.

This is joint work with John Dunn.

Tuesday, 9/26
Title: The Rhodes Semilattice of a Group
Speaker: Michael Gottstein (Binghamton)
Time: 1:15-2:15
Location: WH 100E

The Rhodes semilattice is fundamental in the Krohn–Rhodes complexity theory of finite semigroups. The Rhodes semilattice can be viewed from the perspective of gain graphs. From this perspective the definition is intuitive and simple. In addition there is an immediate, vast generalization to gain-graphic Rhodes semilattices. There are also several natural extensions of the semilattice to non-trivial lattices. I hope to use these lattices to aid in the study of the Rhodes semilattice. Time permitting, I will discuss all of this and more!

Tuesday, 10/3
Title: An Introduction to Strongly Regular and Distance Regular Graphs
Speaker: Quaid Iqbal (Binghamton)
Time: 1:15-2:15
Location: WH 100E

I will introduce strongly regular graphs (SRGs) and distance regular graphs (DRGs), which combine graph theory and matrix theory. First of all, I will give some properties of SRGs and DRGs. Then I will talk about the classification problem, which is to classify distance regular graphs by a fixed parameter.

Tuesday, 10/10
Title: Apexing in Graphs and Its Matroid Analogue
Speaker: Jagdeep Singh
Time: 1:15-2:15
Location: WH 100E

A class of graphs is called hereditary if it is closed under taking induced subgraphs. Its apex class is defined as the class of graphs $G$ that contain a vertex $v$ such that $G-v$ is in the hereditary class. In recent work, Vaidy Sivaraman, Tom Zaslavsky, and I showed that if a hereditary class has finitely many forbidden induced subgraphs, then so does its apex class. I will talk about this result and its matroid analogue.

Tuesday, 10/17
Title: Totally Unimodular Matrices: An Introduction
Speaker: Alireza Salahshoori (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Totally unimodular matrices have very nice properties with respect to solutions of linear equations, linear programming and combinatorial optimization, and matroid theory. I will introduce them and some of the reasons they get attention.

Tuesday, 10/24
Title: Unavoidable Immersions in 4-Edge-Connected Graphs
Speaker: Brittian Qualls (Louisiana State)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

A graph $H$ is called an “immersion” of a graph $G$ if $H$ can be obtained from a subgraph of $G$ by repeated “liftings”, which means replacing two adjacent edges $xy$, $xz$ by one edge $yz$. I discuss results on unavoidable topological minors and their analogous results for immersions. In particular, I discuss my main result (with Guoli Ding): that every sufficiently large 4-edge-connected graph contains a doubled cycle of length $k$, $C_{2,k}$, as an immersion. I will also discuss other results on immersions and possible avenues of further research.

Tuesday, 10/31
Title: Interval Graphs and Representable Complexes
Speaker: Shiyi Ma (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Given a family of $n$ convex sets, we can record their intersection pattern using the “nerve complex”, which is a simplicial complex on the vertex set $[n]$. A simplicial complex $\Delta$ is called “$d$-representable” if it is the nerve of a family of convex sets in $\mathbb{R}^d$, and such a family is called a “$d$-representation” of $\Delta$. For each fixed $d \geq 1$, we obtain asymptotic estimates for the number of $d$-representable simplicial complexes on $n$ vertices as a function of $n$. The case $d = 1$ corresponds to counting interval graphs, which record the nonempty pairwise intersections of $n$ closed intervals on the real line.

This talk is based on the work “Enumeration of interval graphs and $d$-representable complexes”, by Boris Bukh and R. Amzi Jeffs (arXiv:2203.1206).

Tuesday, 11/7
Title: General Certificates of Polytope Non-Realizability
Speaker: Amy Wiebe (U. of British Columbia Okanagan)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/96693197711

A classical question in polytope theory is whether an abstract polytope can be realized as a concrete convex object. Beyond dimension 3, there seems to be no concise answer to this question in general. In specific instances, answering the question in the negative is often done via “final polynomials” introduced by Bokowski and Sturmfels. This method involves finding a polynomial which, based on the structure of a polytope if realizable, must be simultaneously zero and positive, a clear contradiction. The search space for these polynomials is an ideal of Grassmann-Plücker relations, which quickly becomes too large to efficiently search, and in most instances where this technique is used, additional assumptions on the structure of the desired polynomial are necessary.

I will describe how by changing the search space, we are able to use linear programming to exhaustively search for similar polynomial certificates of non-realizability without any assumed structure. We will see that, perhaps surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives and allows us to prove non-realizability of several interesting polytopes.

Tuesday, 11/14
Title: A Matroid Analogue of Chordal Graphs
Speaker: James Dylan Douthitt (Louisiana State)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/93994707886

A graph is chordal if every cycle of length four or more has a chord. In 1961, Dirac proved that a graph is chordal if and only if it can be built from complete graphs by repeated clique unions. I will describe a generalization of Dirac's result to binary matroids.

This talk is based on joint work with James Oxley.

Tuesday, 11/21
No seminar today; it is “Friday”.

Tuesday, 11/28
Title: Excluding a Line from Complex-Representable Matroids
Speaker: Zachary Walsh (Georgia Tech)
Time: 1:15-2:15
Location: WH 100E

I will present a result that highlights the role of gain graphs in the extremal behavior of minor-closed classes of matroids, and then apply this result to determine the extremal behavior for several natural classes of representable matroids. I will assume no knowledge of matroid theory.

This is joint work with Jim Geelen and Peter Nelson.

Tuesday, 12/5
Title: Characteristic Sets of Matroids
Speaker: Marwa Mosallam (Binghamton)
This talk is postponed to the spring.

Friday, 12/8 - change of date (Note special day, time, and room)
Title: Cobiased Graphs
Speaker: Daniel Slilaty (Wright State)
Time: 3:30-4:30
Location: WH 309 and WebEx https://wright.webex.com/meet/daniel.slilaty

Zaslavsky showed that biased graphs effectively characterize single-element extensions and elementary lifts of graphic matroids. I will discuss the dual concept. I will define cobaised graphs and show how they characterize single-element extensions and elementary lifts of cographic matroids. I will also discuss how biased and cobiased graphs reveal more about the structure of their matroids: field representations, orientations, and duality. I will end with some open questions.

SPRING 2023

Tuesday, 1/17
Organizational meeting
Time: 1:30-2:00
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

Tuesday, 1/24
Title: Progress on Projective Rectangles
Speaker: Thomas Zaslavsky (Binghamton)
Time: 1:15-2:15
Location: WH 100E

A projective rectangle is like a projective plane, but narrower. Rigoberto Florez and I have been developing a theory. It's a big project. I will give background and some properties. I will not assume you know what a projective plane is.

Tuesday, 1/31
No seminar today.

Thursday, 2/2: Special lecture
Title: Matroids of Gain Signed Graphs
Speaker: Thomas Zaslavsky (Binghamton)
Time: 3:00-4:00 (EDT)
Location: WH-309 and the Matroid Union Zoom link https://gatech.zoom.us/j/8802082683

For standard affinographic hyperplane arrangements (a.k.a. deformations of the Type A root system arrangement or “braid” arrangement), integral gain graphs give a simpler method to compute the characteristic polynomial, a fundamental invariant. For more general affinographic arrangements (a.k.a. deformations of the Type B root system arrangement), one has to combine gains with signs. How to do this has been a puzzle. The obvious method is to put signs on top of gains. The right method is to put gains on top of signs. Laura Anderson, Ting Su, and I found out how to do this, constructing the natural matroid and the corresponding semimatroid, which latter gives the characteristic polynomial of these more general arrangements when the gain group is the additive group of integers. I will explain some of this. It does get complicated.

Tuesday, 2/7
No seminar today.

Tuesday, 2/14
No seminar today.

Tuesday, 2/21
No seminar today.

Tuesday, 2/28
Title: Current Research on a Weird Poset
Speakers: Michael Gottstein (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Mike will tell us about his work on flat low-dimensional embedding of the simplicial complex of partial partitions.

Tuesday, 3/7
Title: Loose Elements in Some Matroids
Speaker: Jagdeep Singh + Thomas Zaslavsky (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Jagdeep (& Tom) will tell us about loose and nearly loose points in a binary or ternary matroid. Knowledge of matroids will not be necessary.

Tuesday, 3/14
Postponed to next week due to threatening weather.

Tuesday, 3/21
Title: Apex Cographs
Speaker: Jagdeep Singh (Binghamton)
Time: 1:15-2:15
Location: WH 100E

A cograph is a graph that can be built from a vertex by repeated operations of disjoint union and complementation. An apex cograph is a graph that has a vertex $a$ such that deleting $a$ gives a cograph. There is a project to characterize apex cographs, by Vaidy Sivaraman, Tom Zaslavsky, and myself. I will report on it.

Tuesday, 3/28
Title: Cyclic Flats of Matroids and Examples of Why they are Useful
Speaker: Tara Fife (Queen Mary, University of London)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

A matroid is an abstract notion of independence derived from vector spaces, as well as from graphs. Matroids can be characterised in many different ways. Among these is via cyclic flats, that is, the flats of the matroid which are unions of cycles. I will review the definitions of matroids and of cyclic flats. I will explore various classes of matroids and see how cyclic flats were used to characterise them. I will present Joseph Bonin and Anna de Mier's work which characterises matroids via the lattice of cyclic flats. Lastly, we will see how cyclic flats are related to a certain structure used in viewing matroids from an algebraic-geometric perspective.

I am presenting two of my projects, one joint with James Oxley and the other with Felipe Rincon.

Tuesday, 4/11
Title: Classification of Distance Regular Graphs with Diameter $D=3$ and Fixed Eigenvalue
Speaker: Quaid Iqbal (Center for Combinatorics, Nankai University)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

A graph is distance regular if, for any two vertices $v$ and $w$ at distance $d$, the number of vertices at distance $j$ from $v$ and distance $k$ from $w$ depends only on $d$, $j$, and $k$. Distance-regular graphs have very nice eigenvalue properties (e.g., they have $d+1$ distinct eigenvalues) and are a lively topic in spectral graph theory. Distance-regular graphs with diameter 2, which are called strongly regular, have been important in the classification of finite groups.

Given a graph $\Gamma$, the distance-2 graph $\Gamma_2$ is the graph on the same vertices, in which vertices are adjacent if they have distance 2.

I consider the distance-regular graphs $\Gamma$ whose distance-$2$ graphs $\Gamma_2$ are strongly regular. If $\Gamma$ is bipartite, then its distance-$2$ graph is not connected. So, I am interested in the class of non-bipartite distance-regular graphs. First, I will show that the distance-$2$ graph of a non-bipartite distance-regular graph with diameter $3$ and eigenvalue $a_2-c_3$ is strongly regular, and then I will give several kinds of classification of non-bipartite distance-regular graphs with diameter $3$ and eigenvalue $a_2-c_3$ under various conditions, for example when the valency $k$ of $\Gamma$ is at most $2(a_1+1)$, $c_3\leq9$, $a_2\leq 7,$ $\tilde{c} \leq 26$ (where $\tilde{c}$ is the number of common neighbors between any two non-adjacent vertices) and $\theta_{min}(\Gamma) > -3$.

This is joint work with J.H. Koolen, Jongyook Park, and Masood Ur Rehman.

Tuesday, 4/18
Title: Matroids, Polymatroids, and Finite Groups
Speaker: Prairie Wentworth-Nice (Cornell)
Time: 1:15-2:15
Location: WH 100E

First proposed by Whitney in 1935, matroids are combinatorial objects which generalize the notion of linear independence. They have become useful tools in the study of optimization, coding theory, algebraic geometry, and more. One class of matroids that is particularly well studied is the class of representable matroids - those matroids which can be represented over a field. In this talk I look at a generalization of matroids, called subcardinal polymatroids, and discuss a new way to represent these polymatroids by finite groups that is analogous to matroid representability. Using this notion of representability, I classify all matroids representable over non-abelian groups and discuss what is known about matroids representable over abelian groups.

Thursday, 4/20 (Special Event)
Title: Signed Graphs and Gain Graphs
Speaker: Nicholas Lacasse
Time: 1:30-4:00
Location: WH 309 and Zoom https://binghamton.zoom.us/j/95302383985
(Friends of Nick, please show up in person to cheer him on.)

This is Mr. Lacasse's Ph.D. dissertation defense. The dissertation title is “Signed Graphs and Gain Graphs: Packing, Root Systems, Alcoves, and Arrangements”. The examining committee consists of Laura Anderson, Michael Dobbins, Leslie Lander (outside examiner), and Thomas Zaslavsky (advisor and chair).

Abstract

Gain graphs are ordinary graphs equipped with a gain function, which assigns group elements to the arbitrarily oriented edges of the underlying graph. Signed graphs are gain graphs whose gain group is the group of order 2. I will give an overview of my dissertation, which involves four projects where gain graphs are either the object or tool of study.

A signed graph is balanced if all of its circles are positively signed. Negation sets are edge sets in signed graphs whose negation yield a balanced graph. I begin by studying families of pairwise disjoint negation sets in signed graphs. We will discover what such families look like and how to construct them.

Affinographic hyperplane arrangements consist of hyperplanes that have equations of the form $x_i - x_j = k$ where $k$ is an integer. The infinite arrangement $\text{Shi}_n^\infty$ consists of all such hyperplanes. The regions of $\text{Shi}_n^\infty$ are called alcoves and have been used to study the regions of subarrangements of $\text{Shi}_n^\infty$. I will develop a gain-graphic approach to studying alcoves.

Lastly, I will take a close look at affinographic arrangements that consist of hyperplanes whose constant terms are determined by an arithmetic sequence. Such arrangements are called arithmetic arrangements. I will determine the characteristic polynomials of a certain class of arithmetic arrangements. We observe a strange phenomenon where many unions of arithmetic arrangements not only share the same characteristic polynomial but predictably stabilize term-by-term to their common characteristic polynomial.

Tuesday, 4/25
Title: Calculating the Penrose Polynomial from a TQFT
Speaker: Scott Baldridge (Lousiana State)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

The Penrose polynomial of a planar graph, P(G,n), was defined by Roger Penrose in a very important 1971 paper. The polynomial shares many similarities with Kauffman’s bracket and therefore Jones polynomial in knot theory. When the polynomial is evaluated at n=3, it counts the number of 3-edge colorings of a graph (Tait colorings). For n>3, it is nonzero if and only if there is a valid n-face coloring of the graph. Hence, the four-color theorem is true if all bridgeless planar graphs G satisfy P(G,4)>0. Unfortunately, the Penrose polynomial is too weak an invariant to be able to prove the four-color theorem directly. But two categorifications of it, i.e., the bigraded n-color homology and filtered n-color homology, are far stronger invariants that do appear to have the power to prove it.

I will introduce these homology theories and generalize the Penrose polynomial to the “total color polynomial,” which is a new abstract graph invariant when the graph is trivalent. If the graph also has trivial automorphism group, then the total color polynomial evaluated at n is the sum of the counts of n-face colorings on all distinct ribbon graphs that can be formed from the graph. For topologists, a ribbon graph is the (neighborhood of the) 1-skeleton of a CW complex of a smooth closed surface.

To get to the total color polynomial, one thinks of the different proper n-face colorings on the 2-cells of the CW complexes associated to a graph as a system of states as in physics. This state system is introduced through a complicated “TQFT” (a topological quantum field theory, which will be suppressed for this talk), but which then leads to the homology theories above. In some sense they are analogous to Khovanov homology and Lee homology in knot theory. The Penrose polynomial is then the Euler characteristic of the bigraded homology (think “Khovanov homology”) and the total color polynomial is the Poincaré polynomial of the filtered n-color homology (think “Lee homology”). These new homologies and the total color polynomial are far stronger invariants than the Penrose polynomial, and if we have time, I will discuss how they relate to the four-color theorem.

This talk will be hands-on and the ideas will be explained through the calculation of easy examples! My goal is to attract mathematicians to this area by making the ideas as intuitive as possible. Topologists and representation theorists are encouraged to attend also—these homologies sit at the intersection of topology, representation theory, and graph theory.

This is joint work with Ben McCarty.

Tuesday, 5/2
No seminar: today is officially Friday.

FALL 2022

Tuesday, 8/23
Organizational meeting
Time: 2:00-2:30 (Note special time)
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

Tuesday, 8/30
Title: What I'm Doing
Speakers: Stefan Viola, Tara Koskulitz, Jagdeep Singh (Binghamtons)
Time: 1:15-2:15
Location: WH 100E

Tuesday, 9/6
No seminar: it is “Monday”.

Tuesday, 9/13
Title: Perfectly Matchable Set Polynomials and an Application to Ehrhart Theory
Speaker: Robert Davis (Colgate)
Time: 1:15-2:15
Location: WH 100E

A subset S of vertices of a graph G is called a perfectly matchable set of G if the subgraph induced by S contains a perfect matching. The perfectly matchable set polynomial of G, first made explicit by Ohsugi and Tsuchiya, is the (ordinary) generating function p(G; z) for the number of perfectly matchable sets of G. I will compare p(G; z) to the classical matching polynomial and provide explicit recurrences for computing p(G; z) for an arbitrary (simple) graph. I will use these to compute the Ehrhart h*-polynomials for certain lattice polytopes, which was the original motivation for this work. Namely, I show that p(G; z) is the h*-polynomial for certain classes of stable set polytopes, whose vertices correspond to stable sets of G.

This is joint work with Florian Kohl.

Tuesday, 9/20
Title: From Cographs to 2-Cographs and Comatroids
Speaker: Jagdeep Singh (Binghamton)
Time: 1:15-2:15
Location: WH 100E

A graph all of whose connected induced subgraphs have a disconnected complement is called a cograph. Such graphs can be recursively built from a single vertex via complementation and disjoint union and, therefore, are called complement-reducible graphs as well. Another characterization of cographs is that they are precisely the graphs that do not have an induced path of length three. In this talk, I replace connectivity with 2-connectivity and consider this natural generalization, called 2-cographs, of the class of cographs. I will talk about the corresponding results for the class of 2-cographs and an analogue of cographs for matroids.

This is joint work with James Oxley.

Tuesday, 9/27
No seminar; it is New Year's Day (Rosh Hashonah).

Tuesday, 10/4
No seminar; it is Yom Kippur Eve.

Tuesday, 10/11
Title: Whitney Numbers
Speaker: Thomas Zaslavsky (Binghamton)
Time: 1:15-2:15
Location: WH 100E

For each finite group $G$ and each rank $n$ there is a Dowling lattice, a geometric lattice, $Q_n(G)$. The coefficients of the characteristic polynomial of this lattice are the Whitney numbers of the first kind of $Q_n(G)$, $w_i$ for $i=0,1,\ldots,n$. They depend only on $n$ and the order of the group, $|G|$. It is easy to see that $w_i(|G|)$ is a polynomial function of $|G|$. I will discuss the not-so-easy details, such as the degree and the coefficients in the polynomial $w_i(q)$, for an extensive generalization of the Dowling lattices that is obtained by using gain graphs. I will explain almost all of this in the talk.

Tuesday, 10/18
Title: Monotonic Linear Models
Speaker: Laura Anderson (Binghamton)
Time: 1:15-2:15
Location: WH 100E

I'll discuss new methods for data analysis related to psychological models.

The models in question are of the form $y=f(Ax)$, where $x$ is a vector of independent variables that cannot be measured, $y$ is a vector of data to be measured in an experiment, $A$ is a matrix given by the theory being tested, and $f$ is an unknown coordinate-wise increasing function. (The first part of the talk will explore how such a model arises.)

Assume that you have run the experiment and have a bunch of vectors $y$ of data. How do you characterize how well your data fits the model?

If one assumes $f$ to be the identity function, then the answer is elementary. This explains why people frequently make this assumption, even when it has no theoretical justification. Answering the question without making assumptions about the form of $f$ leads one into combinatorics. A particularly knotty issue is the question of error: how does one measure how far a data point $y$ is from satisfying combinatorial conditions? I'll discuss joint work with John Dunn addressing this question.

Tuesday, 10/25
Title: Inscribable Order Types
Speaker: Michael Dobbins (Binghamton)
Time: 1:15-2:15
Location: WH 329 (Note change of room.)

We call an order type of points in the plane “inscribable” if it can be realized by a point configuration where all extreme points are on a circle. I will show that every simple order type with either at most 2 interior points or at most 5 extreme points is inscribable. Then, I will present an uninscribable configuration with 3 interior and 6 extreme points, which we call the non-Pascal configuration. Then, I will construct an infinite family of minimally uninscribable order types using properties of Möbius transformations.

This is joint work with Seunghun Lee.

Tuesday, 11/1
Speaker: No speaker today.

Tuesday, 11/8
Speaker: No speaker today.
It's Election Day: vote (if a U.S. citizen).

Tuesday, 11/15
Title: 1-Skeleton Posets of Bruhat Interval Polytopes
Speaker: Christian Gaetz (Cornell)
Time: 1:15-2:15 (followed by an Algebra Seminar talk; see below)
Location: WH 100E

Bruhat interval polytopes are combinatorially interesting polytopes arising from total positivity and from certain toric varieties. I study the 1-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. Interestingly, these posets turn out to be lattices and the polytopes, despite not necessarily being simple, have interesting h-vectors. I will give a criterion for determining when these polytopes are simple, or equivalently when generic torus orbit closures in Schubert varieties are smooth, solving a conjecture of Lee and Masuda.

Algebra Seminar
Title: Stable Characters from Permutation Patterns
Speaker: Christian Gaetz (Cornell)
Time: 2:50-3:50
Location: WH 100E

Christopher Ryba, Laura Pierson, and I study the expected value (and higher moments) of the number of occurrences of a fixed permutation pattern on conjugacy classes of the symmetric group $S_n$. We prove that this virtual character stabilizes as n grows, so that there is a single polynomial computing these moments on any conjugacy class of any symmetric group. Our proof appears to be the first application of partition algebras to the study of permutation patterns. I’ll also discuss partial progress towards a conjecture on when these virtual characters are genuine characters.

Tuesday, 11/22
Title: Random Currents and Continuity of Ising Model’s Spontaneous Magnetization
Speaker: Gang Zhou (Binghamton)
Time: 1:15-2:15
Location: WH 100E

I will present the paper “Random currents and continuity of Ising model’s spontaneous magnetization” by M. Aizenman, H. Duminil-Copin and V. Sidoravicius (2015).

In the paper they considered three-dimensional ferromagnetic Ising model. It is known that at the high temperature, the system is at disorder; at the low temperature, the system exhibits ferromagnetic order, or magnetization. They proved that at the critical temperature, the magnetization is continuous, which was a long standing conjecture.

A crucial technique is the so-called switching lemma. It establishes a bijection between undirected graphs generated by the random current representation. In many important papers this was used, including the ones helping Hugo Duminil-Copin to win a Fields medal in 2022. However this technique does not work for the other spin models, for example, XY model or most of the quantum models.

This is the same talk I gave at the analysis seminar, but for the people who could not attend it for various reasons.

Tuesday, 11/29
Title: A New Length Estimate for Curve-Shortening Flow and Low Regularity Initial Data
Speaker: Shiyi Ma (Binghamton)
Time: 1:15-3:30 (Note longer duration.)
Location: WH 329 (Note special room.)

Part 1: We want to understand how Jordan curves evolve by curve-shortening flow in $\mathbb{R}^2$. Short-time existence for smooth initial data has been proved by Gage and Hamilton. When we are trying to show that curve-shortening flow is able to smooth non-smooth initial data, one of the major obstacles is that analytic estimates are difficult to control when the lengths of any approximating sequence are unbounded. I will be discussing work by Joseph Lauer in which he introduces a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve-shortening flow. The length estimates we obtain are used to prove results about the level set flow in $\mathbb{R}^2$.

Part 2: I will show the smoothness of level set flow. If $K$ is locally connected, connected and compact, then the level set flow of $K$ either vanishes instantly, fattens instantly or instantly becomes a smooth closed curve. If the compact set in question is a Jordan curve γ, then γ instantly becomes either a smooth curve or an annular region with smooth boundary.

This talk will be the candidacy exam of Shiyi Ma. The examining committee consists of Michael Dobbins (chair), Thomas Zaslavsky, and Gang Zhou.

Tuesday, 12/6
Title:
Speaker:
Time: 1:15-2:15
Location: WH 100E

SUMMER 2022

Friday, 7/1
Title: Geometry of Matroids and Hyperplane Arrangements
Speaker: Jaeho Shin (Korea Institute for Advanced Study)
Time: 3:30-4:30 p.m. Note special time
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

There is a trinity relationship between hyperplane arrangements, matroids and convex polytopes. There are at least three different starting points to understand the relationship. In this talk, I will take a path from the hyperplane arrangements and explain as much as time allows.

Thursday, 7/14
Titles:
Part 1. Combinatorial Obstructions to the Lifting of Link Diagrams
Part 2. Cycling in Link Diagrams and Noneuclidean Oriented Matroids
Speaker: Tara Koskulitz (Binghamton)
Time: 1:00–3:15
Location: WH 100E

Part 1: Consider a line arrangement in $\mathbb{R}^3$. When we draw it in the plane, we draw an arrangement of lines together with some indication of the above-below relations at each point where two lines intersect. This realizable situation inspires the general definition of a link diagram: a pair consisting of a line arrangement in $\mathbb{R}^2$ with a function giving “above-below” relations at each intersection. We can then ask whether or not a general link diagram is liftable to an actual arrangement of lines in $\mathbb{R}^3$. I will be discussing work by Jürgen Richter-Gebert in which he introduces methods for using oriented matroids to solve these liftability problems.

Part 2: A particular type of obstruction to lifting occurs when the link diagram contains a nondegenerate cycle. In this case, the associated (partial) oriented matroid is noneuclidean.

This is Ms. Koskulitz's candidacy exam. All are invited. The examining committee consists of Laura Anderson (chair), Michael Dobbins, and Thomas Zaslavsky.

Wednesday, 7/20
Title: Mutations of Mystic Monoliths
Speaker: Christopher Eppolito (Binghamton)
Time: 1:00–3:15 (with a break)
Location: WH 100E

A Pythagorean hyperplane arrangement is determined by a configuration of reference points in affine space and a real gain graph. The combinatorics of such arrangements is thus related to both the geometry of these points and the combinatorics of the gain graph. Previous work in this area by T. Zaslavsky determined region-counting formulas for the configuration-generic Pythagorean arrangements, i.e., those with stable intersection pattern under perturbation of the reference points.

In recent work, I proved two results on Pythagorean hyperplane arrangements. The first theorem constructs the intersection pattern of arbitrary Pythagorean arrangements, from which region-counting formulas follow via T. Zaslavsky's thesis. The main tool is an auxiliary hyperplane arrangement in the real edge space of the graph which incorporates the geometry of a fixed set of reference points. The flats of this arrangement are in bijection with the possible intersection patterns. The second theorem is an extension result for configuration-generic arrangements. The main gadget here is a new operation on gain graphs. Fixing a gain graph, we use this operation to construct a finite set of spheres and hyperplanes in affine space which determine the locus of the extra points that result in a configuration-nongeneric arrangement.

This talk begins with an overview of my dissertation. Following this I discuss the results on Pythagorean arrangements in detail. I include many pictures (the true purpose of the talk).

This is Mr. Eppolito's dissertation defense. All are invited. The examining committee consists of Laura Anderson (co-advisor), Michael Dobbins, Leslie Lander (outside examiner), and Thomas Zaslavsky (chair and co-advisor).

Thursday, 7/21
Titles:
Part 1. Geometric Algebra for Matroids
Part 2. Foundation of a Matroid (brief summary)
Speaker: Stefan Viola (Binghamton)
Time: 1:00–3:00
Location: WH 100E

Part 1: In 1989 Dress and Wenzel showed that for any matroid $M$ on a finite set $E$, a certain abelian group $T_M^H$ (called the extended Tutte group of $M$) can be canonically associated with $M$. The extended Tutte group has several cryptomorphic characterizations, but I will focus only on the characterization given by the hyperplanes of $M$. I will show that the cross-ratio, which is an important tool in classical projective geometry, extends naturally to matroids via the theory of Tutte groups. I will give a non-realizability condition in terms of cross-ratios and a Tutte-group theoretic proof of Pappus's theorem.

This talk is based on the paper “On Combinatorial and Projective Geometry” by Andreas Dress and Walter Wenzel (1990).

Part 2 (will be a short summary): In 2020 Baker and Lorscheid introduced the foundation of a matroid, which is an algebraic invariant that classifies all realizations of the matroid up to rescaling. I will give a presentation for the foundation of a matroid in terms of generators and relations, where the generators are the universal cross-ratios of the matroid and all relations between universal cross-ratios are inherited from embedded minors having at most 7 elements.

This talk is based on the paper “Foundations of Matroids Part 1: Matroids without Large Uniform Minors” by Matthew Baker and Oliver Lorscheid (2020).

This is Mr. Viola's candidacy exam. All are invited. The examining committee consists of Laura Anderson (chair), Michael Dobbins, and Thomas Zaslavsky.

SPRING 2022

Tuesday, 1/25
Title: Organizational meeting
Speaker: Everyone and anyone
Time: 1:30-2:15 (Note special starting time.)
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

Tuesday, 2/1
Title: Unimodular Triangulations of Sufficiently Large Dilations
Speaker: Gaku Liu (U. of Washington)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985Dilations

An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope in $\mathbb R^d$ is a triangulation in which all simplices are integral with volume $1/d!$. A classic result of Kempf, Mumford, and Waterman states that for every integral polytope $P$, there exists a positive integer $c$ such that $cP$ has a unimodular triangulation. I strengthen this result by showing that for every integral polytope $P$, there exists $c$ such that for every positive integer $c' \ge c$, $c'P$ admits a unimodular triangulation.

Tuesday, 2/8
Title: Open Problems
Speaker: Seunghun Lee et al. (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Tuesday, 2/15
Title: Sweep Oriented Matroids
Speaker: Arnau Padrol (Institut de Mathématiques de Jussieu)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

A sweep of a point configuration is an ordered partition induced by a linear functional. The set of orderings obtained this way is highly structured: isomorphic to the face lattice of a convex polytope, the sweep polytope. In the plane, they were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations, but a high dimensional generalization was missing. Mimicking the fact that sweep polytopes of point configurations are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. I will present sweep oriented matroids, their connection with Dilworth truncations and the generalized Baues problem for cellular strings, and many open questions.

This is based on joint work with Eva Philippe.

Tuesday, 2/22
Title: A Dr. Strange Partial Ordering of Partial Partitions
Speaker: Mike Gottstein (Binghamton)
Time: 1:15-2:15
Location: WH 100E

I will introduce the poset of partial set partitions ordered by set inclusion, as opposed to the typical ordering by refinement. The goal is to see what the homology of this poset is. I will first argue by an elementary approach, and then introduce the concept of shellability to see how this gives the same solution only in a much wider scope.

Tuesday, 3/1
Title: A New Matroid Lift Construction and an Application to Gain Graphs
Speaker: Zach Walsh (Louisiana State)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

Crapo constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. I generalize by constructing a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of circuits of $M$. I conjecture that every lift of $M$ arises via this construction.

I apply this result to gain graphs, generalizing a construction of Zaslavsky. Given a graph $G$ with gains (edges labeled invertibly by a group), Zaslavsky’s lift matroid $K$ is an elementary lift of the graphic matroid $M(G)$ that respects the gains; specifically, the cycles of $G$ that are circuits of $K$ coincide with the cycles that have neutral gain. For $k \geq 2$, when does there exist a rank-$k$ lift of $M(G)$ that respects the gains? For abelian groups, I show that such a matroid exists if and only if the group is the additive group of a non-prime finite field.

Tuesday, 3/8
Title: The Inverse Kakeya Problem
Speaker: Michael Dobbins (Binghamton)
Time: 1:15-2:15
Location: WH 100E

I show that the largest convex body that can be placed inside a given convex body $Q$ in $\mathbb{R}^d$ in every orientation is the largest inscribed ball of $Q$. This is true for both largest volume and for largest surface area. Furthermore, the ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.

This is joint work with Sergio Cabello and Otfried Cheong.

Tuesday, 3/15
No seminar; happy holiday!

Tuesday, 3/22
Title: Mix-Ups of Matrices and Graphs
Speaker: Thomas Zaslavsky (Binghamton)
Time: 1:15-2:15
Location: WH 100E

There are many kinds of graph, from simple graphs to complex unit gain graphs and including directed graphs, oriented graphs, signed graphs, mixed graphs, mixed-up graphs, and whatever. Each one has an adjacency matrix—or two or three—that describes the graph. I will survey what little I know about all those matrices and what people are studying about them.

Tuesday, 3/29
Title: Coloring Graphs and Their Complements
Speaker: Peter Maceli (Ithaca)
Time: 1:15-2:15
Location: WH 100E

Nordhaus and Gaddum showed that for any graph the sum of its chromatic number together with the chromatic number of its complement is at most one more than the number of vertices in the graph. The class of graphs which satisfy this upper bound with equality have long been understood; however, not much beyond this initial case is known in terms of characterizing graphs via the sum of complementary chromatic numbers. I will discuss how adopting a more structural approach to this general problem leads to an interesting method of graph decomposition, which in turn allows one to generalize and extend several previous results.

Tuesday, 4/5
Title: Higher Nerves and Applications
Speaker: Hai Long Dao (Kansas)
Time: 1:15-2:15
Location: WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

I introduce a generalized version of the nerve complexes. I show that when applied to the Stanley–Reisner scheme of a simplicial complex, these higher nerve complexes can be used to compute important homological and combinatorial invariants such as depth, the $h$-vector, and the $f$-vector.

Tuesday, 4/12
Special Event
Discrete & Computational Geometry Day
In Memory of Eli Goodman and Ricky Pollack

Location: Zoom https://springer.zoom.us/j/6440052748
Time: 12:30-4:05

Program
12:30 Janos Pach (Renyi Institute): Welcome & Introduction
12:40 Andreas Holmsen (KAIST): An allowable feast
13:15 Micha Sharir (Tel Aviv University): Polynomial partitioning: The hammer and some (recent algorithmic) nails
13:50 Esther Ezra (Bar Ilan University): Recent developments on intersection searching
14:25 Xavier Goaoc (Loria, Nancy): Some questions on order types
15:00 Andrew Suk (UC San Diego): Unavoidable patterns in simple topological graphs
15:35 Sylvain Cappell (Courant Institute): Mesh matrices of graphs, of simplicial complexes and of matroids and the significance of their eigenvalues

Tuesday, 4/19
No seminar; it's “Monday”.

Tuesday, 4/26
Title: What I Remember From Saturday's Discrete Math Day at Colgate
Speaker: Thomas Zaslavsky (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Very Abstract: There were interesting if weird talks, and I spoke with several people about various kinds of math.

Tuesday, 5/3
No seminar. We are closed for repairs until next week.

Tuesday, 5/10
Title: Projectivities in Simplicial Complexes and Balanced Spheres
Speaker: Seunghun Lee (Binghamton)
Time: 1:15-2:15
Location: This will be a remote talk. View in WH 100E and Zoom https://binghamton.zoom.us/j/95302383985

In his paper “Projectivities in simplicial complexes and colorings of simple polytopes”, Joswig studied the group of projectivities of a simplicial complex, and applied it to obtain equivalent conditions to define a balanced sphere, a d-dimensional simplicial sphere whose graph is properly (d+1)-colorable. Main results and proof ideas will be introduced in this talk.

Wednesday, 5/11 (Special day)
Talk 1: Homeomorphism Type of Combinatorial Grassmannian and Flag Manifold
Talk 2: Space of Flattenings of Spheres and Homotopy Type of Combinatorial Grassmannian Bundles
Speaker: Olakunle Abawonse (Binghamton)
Time: 3:30 p.m.
Location: WH 100E and on Zoom https://binghamton.zoom.us/j/95738127171

This is Mr. Abawonse's dissertation defense. The dissertation committee consists of Laura Anderson (chair), Michael Dobbins, Cary Malkiewich, and Florian Frick (outside examiner).

The dissertation defense is open to the public. All are invited to attend.

Abstract:

I will show that the geometric realizations of the poset of rank two oriented matroids and the poset of flags of rank one and rank two oriented matroids are homeomorphic to their corresponding Grassmannian counterparts. The proof will involve shellings of intervals in the two posets and face collapse on the boundary of a polytope.

I will also establish that the space of flattenings of some simplicial spheres, like a simplicial one-sphere and the join of the boundaries of a 1-simplex and a k-simplex, is homotopy equivalent to an orthogonal group.

Lastly, I will establish sufficient combinatorial conditions under which there is a fixing cycle associated to a triangulation of a smooth manifold. A fixing cycle is a homology class analogous to the fundamental class of a Grassmannian bundle. In the course of this proof, I will consider the concept of a poset of oriented matroid charts as a combinatorial abstraction to the space of flattenings of spheres.

Thursday, 5/12 (Special day)
Title: On the Topology of Corank 1 Tropical Phased Matroids
Speaker: Ulysses Alvarez (Binghamton)
Time: 5:00 p.m.
Location: WH G002 and on Zoom, https://binghamton.zoom.us/j/97930954632

This is Mr. Alvarez's dissertation defense. The dissertation committee consists of Laura Anderson (chair), Michael Dobbins, Ross Geoghegan (co-advisor), and Les Lander (outside examiner).

The dissertation defense is open to the public. All are welcome to attend.

For each topological poset X we can associate a topological space we call the topological order complex of X. Past work done by Ross Geoghegan and me shows that the set of nonzero covectors of a matroid over the tropical phase hyperfield, which can be given a topological poset structure, has the same weak homotopy type as its associated order complex. Given such a matroid of corank (i.e., dual rank, or nullity) 1, the work in this dissertation shows that the associated order complex is homeomorphic to a sphere by equipping the order complex with a regular cell decomposition. Thus the set of nonzero covectors of a corank 1 matroid over the tropical phase hyperfield is weakly homotopy equivalent to a sphere.

FALL 2021

Tuesday, 8/31
Title: Polytopes Arising From {1,3}-Graphs: Ehrhart Quasi-Polynomial and Scissors Congruence Phenomenon
Speaker: Jorge L. Ramírez Alfonsín (Montpellier)
Time: 1:15-2:15
Location: Zoom https://binghamton.zoom.us/j/95302383985
in WH 100E and wherever you want to Zoom.

Liu and Osserman introduced a family of polytopes, naturally associated to graphs whose vertices have degrees one and three, and studied their Ehrhart quasi-polynomials. The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. After a gentle introduction to Ehrhart theory, I present a proof of this conjecture for the class of polytopes mentioned above. The key ingredient in the proof is the nearest neighbor interchange on graphs.

I also present some results on the period of the Ehrhart quasi-polynomial as well as some nice geometric and combinatorial properties of such polytopes.

This is joint work with Cristina G. Fernandes, José C. de Pina, and Sinai Robins.

Tuesday, 9/7
No seminar (holiday).

Tuesday, 9/14
Title: What I've Been Thinking About, Part I
Speakers: Uly Alvarez (Topological posets), Seunghun Lee (Geometric hypergraph transversals)
Time: 1:15-2:15
Location: WH 100E

Tuesday, 9/21
Title: What I've Been Thinking About, Part II
Speakers: Chris Eppolito (Pythagorean hyperplanes), Mike Gottstein (Partition containment), Tom Zaslavsky (Projective rectangles)
Time: 1:15-2:15
Location: WH 100E

Thursday, 9/23, in the Geometry/Topology Seminar
Title: A Strong Equivariant Deformation Retraction from the Homeomorphism Group of the Projective Plane to the Special Orthogonal Group
Speaker: Michael Dobbins (Binghamton)
Time: 2:50-3:50
Location: WH 100E

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.

Tuesday, 9/28
Title: Recent Work of Baker and Lorscheid on “Foundations of a Matroid”
Speaker: Laura Anderson (Binghamton)
Time: 1:15-2:15
Location: WH 100E

Thursday, 9/30, in the Geometry/Topology Seminar
Title: The topology of a corank-1 matroid over Φ
Speaker: Ulysses Alvarez (Binghamton)
Time: 2:50-3:50
Location: WH 100E

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.

AND: Thursday, 9/30, UCLA Combinatorics Seminar
Title: Log-Concave Inequalities for Posets
Speaker: Swee Hong Chan (UCLA)
Time: 2:00-3:00 PDT, 5:00-6:00 EDT
URL for livestream access from UCLA: https://www.math.ucla.edu/~galashin/ucla_comb_sem.html

The study of log-concave inequalities for combinatorial objects has seen much progress in recent years. One such progress is the solution to the strongest form of Mason's conjecture (independently by Anari et al. and Brándën–Huh) that the $f$-vectors of matroid independence complexes are ultra-log-concave [i.e., binomially concave–T.Z.]. In this talk, I discuss a new proof of this result through linear algebra and discuss generalizations to greedoids and posets. This is a joint work with Igor Pak.

The talk is aimed at a general audience.

Tuesday, 10/5
Title: Ehrhart Theory of Rank-Two Matroids
Speaker: Benjamin Schröter (KTH Royal Institute of Technology)
Time: 1:15-2:15
Location: Zoom https://binghamton.zoom.us/j/95302383985
in WH 100E and wherever you want to Zoom.

There are many questions that are equivalent to the enumeration of lattice points in convex sets. Ehrhart theory is the systematic study of lattice point counting in dilations of polytopes. Over a decade ago De Loera, Haws and Köppe conjectured that the lattice point enumerator of dilations of matroid basis polytopes is a polynomial with positive coefficients. This intensively studied conjecture has recently been disproved in all ranks greater than or equal to three. However, the questions of what characterizes these polynomials remain wide open.

In this talk I will report on my work, with Katharina Jochemko and Luis Ferroni, in which we complete the picture on Ehrhart polynomials of matroid basis polytopes by showing that they have indeed only positive coefficients in low rank. Moreover, we also prove that the closely related $h^*$-polynomials of sparse paving matroids of rank two are real-rooted, which implies that their coefficients form log-concave and unimodal sequences.

Tuesday, 10/12
Title: Balanced Matchings and Admissible Division
Speaker: Joseph Briggs (Auburn University)
Time: 1:15-2:15
Location: Zoom https://binghamton.zoom.us/j/95302383985
in WH 100E and wherever you want to Zoom.

The KKM theorem offers a sufficient topological condition for a collection of $d$ closed sets covering the $(n-1)$-dimensional simplex to have a common point in their intersection. It is a continuous version of Sperner's Lemma, and it has important relations to game theory. For example, a classical result of Gale, Woodall and Stromquist says that it is always possible to divide a cake among $n$ hungry players so that each is content with their own assigned piece.

But unfortunately, the same desirable situation is no longer true once there is more than one cake. I will introduce this multiple-cake division problem: If we allow some pieces of cake to be left uneaten (or some players left hungry), how many players can still be simultaneously placated? I will also discuss its relation to an extremal parameter involving matchings in fractionally balanced hypergraphs, which I will use to answer some questions of this form.

Tuesday, 10/19
Title: Quiver Representations Over the Field with One Element
Speaker: Jaiung Jun (New Paltz)
Time: 1:15-2:15
Location: Zoom https://binghamton.zoom.us/j/95302383985
in WH 100E and wherever you want to Zoom.

A quiver is a directed graph, and a representation of a quiver assigns a vector space to each vertex and a linear map to each arrow. Quiver representations arise naturally from problems in the representation theory of associative algebras. Instead of vector spaces and linear maps, one may consider a combinatorial model for quiver representations by replacing vector spaces and linear maps with “vector spaces over the field with one element $F_1$” and “$F_1$-linear maps”. I will introduce several aspects of quiver representations over the field with one element.

This is joint work with Jaehoon Kim and Alex Sistko.

Tuesday, 10/26
Title: Covering Numbers of Graphs
Speaker: Casey Donoven (Montana State-Northern)
Time: 1:15-2:15
Location: WH 100E (in person)

Given a structure $A$ and a class $C$ of substructures, the covering number of $A$ with respect to $C$ is the minimum number of substructures whose union is $A$. Covering numbers have been explored in various disciplines, such as algebra (for groups, rings, loops, semigroups, etc.) and graph theory (biparticity, etc.). I will discuss several covering numbers for graphs, including: covering graphs with complete subgraphs, covering bipartite graphs with complete bipartite subgraphs, and covering digraphs with dicuts. For each, I will establish a theorem describing the covering number obeying the following meta-theorem: there exists a sequence of graphs $\Delta_n$ such that the covering number of $\Gamma$ is the minimum $n$ such that there is an appropriate function from $\Gamma$ into $\Delta_n$.

Tuesday, 11/2
Title: Pairs of Graphs With the Same Even Cycles
Speaker: Bertrand Guenin (Waterloo)
Time: 1:15-2:15
Location: Zoom https://binghamton.zoom.us/j/95302383985
in WH 100E and wherever you want to Zoom.

Whitney proved that if two 3-connected graphs have the same set of cycles (or equivalently, the same set of cuts) then both graphs must be the same. I characterize when two 4-connected signed graphs have the same set of even cycles, and I characterize when two 4-connected grafts have the same set of even cuts.

This is joint work with Cheolwon Heo and Irene Pivotto.

[Notes: “Even” means positive. A graft is a kind of dual of a signed graph. – TZ]

Tuesday, 11/9
No seminar today.

Tuesday, 11/16
Title: Point Arrangements and Oriented Matroids from Biological Data
Speaker: Caitlin Leinkamper (Penn State)
Time: 1:15-2:15
Location: WH 100E (in person)

Determining the rank of a matrix derived from data is a fundamental problem in many fields of biology. However, it is often impossible to measure the true quantity of interest, and we must instead measure a proxy value which has a monotone relationship with the true value. This motivates the following definition: the underlying rank of a matrix A is the minimum rank d such that there is a rank-d matrix B whose entries are in the same order as the entries of A. I introduce a variety of strategies for estimating underlying rank. Using results about random polytopes, I give techniques for estimating the underlying ranks of random matrices which I use to estimate the dimensionality of neural activity in zebrafish. Next, I use Radon's theorem to derive a basic lower bound for underlying rank. I show that underlying rank can exceed this bound using examples derived from oriented matroids and allowable sequences. I show that for d≥2, determining whether a matrix has underlying rank at most d is complete for the existential theory of the reals, and therefore NP hard. However, for d=2, one can solve a combinatorial relaxation of this problem in polynomial time.

Tuesday, 11/23
No seminar today.

Tuesday, 11/30
Title:
Speaker:
Time: 1:15-2:15
Location:

Tuesday, 12/7
Title:
Speaker:
Time: 1:15-2:15
Location:

SUMMER 2021

Friday, 7/9 (special day)
Title: Homotopy Type of the Independence Complexes of Forests
Speaker: Michael Gottstein (Binghamton)
Time: 1:00-3:00

Zoom meeting https://binghamton.zoom.us/j/95302383985

Abstract: I first will show the independence complexes of forests are instances of a class of simplicial complexes for which I show that each member is contractible or homotopy equivalent to a sphere. I then will provide an inductive method of detecting the contractibility of the complex or the dimension of the sphere associated to a forest. Time permitting I will mention other examples of complexes in this class. The talk is based on two papers: “The topology of the independence complex” by Richard Ehrenborg and Gabor Hetyei (2006) and “Homotopy types of independence complexes of forests” by Kazuhiro Kawamura (2010).

This is Mr. Gottstein's examination for admission to candidacy. The examining committee consists of Laura Anderson, Michael Dobbins, and Thomas Zaslavsky (chair).

All interested persons are welcome to participate by Zoom.

SPRING 2021

Tuesday, 2/16
Organizational meeting
Time: 1:15-2:15

Zoom meeting id: 953 0238 3985

Tuesday, 2/23
Title: What I've Been Doing, Part I
Speakers: Thomas Zaslavsky, James West, Michael Dobbins
Time: 1:15-2:15

Zoom meeting id: 953 0238 3985

Tuesday, 3/2
Title: What I've Been Doing, Part II
Speakers: Olakunle Abawonse, Seunghun Lee
Time: 1:15-2:15

Zoom meeting id: 953 0238 3985

Tuesday, 3/9
Title: None
Speakers: None
Time: 1:15-2:15

Zoom meeting https://binghamton.zoom.us/j/95302383985

Tuesday, 3/16
Title: The Erdös–Faber–Lovász Conjecture and Related Results
Speaker: Dong-yeap Kang (Birmingham [U.K.])
Time: 1:15-2:15

A hypergraph is linear if every pair of two distinct edges shares at most one vertex. A longstanding conjecture by Erdös, Faber, and Lovász in 1972, states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. I will discuss the ideas to prove the conjecture for all large $n$.

This is joint work with Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus.

Zoom meeting https://binghamton.zoom.us/j/95302383985

Tuesday, 3/23
Title: Chi-Boundedness
Speaker: Peter Maceli (Ithaca College)
Time: 1:15-2:15

Zoom meeting https://binghamton.zoom.us/j/95302383985

If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? In the early 1980’s András Gyárfás made a number of challenging conjectures about this. I will give a survey of how these questions seek to generalize the class of perfect graphs, along with some recent results.

Tuesday, 3/30
Title: Quantitative Helly Theorems
Speaker: Pablo Soberón (Baruch)
Time: 1:15-2:15

Given a family of convex sets in R^d, how do we know that their intersection has a large volume or a large diameter? A large family of results in combinatorial geometry, called Helly-type theorems, characterize families of convex sets whose intersections are not empty. I will describe how some bootstrapping arguments allow us to extend classic results, to describe when the intersection of a family of convex sets in R^d is quantifiably large.

The work presented in this talk was done in collaboration with Travis Dillon, Jack Messina, Sherry Sarkar, and Alexander Xue.

Zoom meeting https://binghamton.zoom.us/j/95302383985

Thursday, 4/1 in the Topology Seminar
Title: Gelfand and MacPherson's Combinatorial Formula for Pontrjagin Classes, Part I: The Topology
Speaker: Olakunle Abawonse
Time: 2:50-3:50
Zoom meeting TBA

See the abstract for the next talk, 4/6.

Tuesday, 4/6
Title: Gelfand and MacPherson's Combinatorial Formula for Pontrjagin Classes, Part II: The Combinatorics
Speaker: Laura Anderson
Time: 1:15-2:15

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 Švarc(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. 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 I (in the Topology Seminar on 4/1). The second part is a combinatorial model for differential manifolds that admits a combinatorialization of this Chern–Weil theory. We'll discuss this in Part II (in the Combinatorics Seminar on 4/6).

Either talk should be of interest independently of the other.

Zoom meeting https://binghamton.zoom.us/j/95302383985

Tuesday, 4/13
Title: Oriented Matroids from Triangulations of Products of Simplices
Speaker: Chi Ho Yuen (Brown)
Time: 1:15-2:15

Marcel Celaya, Georg Loho, and I introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. We also derive a topological representation of the oriented matroid using a variant of Viros patchworking and we describe the extension to matroids over hyperfields.

Zoom meeting https://binghamton.zoom.us/j/95302383985

Tuesday, 4/20
Title: Fork-Free Graphs and Perfect Divisibility
Speaker: Vaidyanathan Sivaraman (Mississippi State)
Time: 1:15-2:15

The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. What is the structure of graphs not containing the fork as an induced subgraph? In particular, can we bound the chromatic number of such a graph in terms of its clique number? Such a chi-bounding function is known but it is not known whether a polynomial function would suffice. I conjectured that such graphs satisfy a strong property called ''perfect divisibility”, which in turn will yield a quadratic chi-bounding function. I will discuss results proving the conjecture in some subclasses of the class of fork-free graphs.

This is joint work with T. Karthick and Jenny Kaufmann.

Zoom meeting https://binghamton.zoom.us/j/95302383985

Tuesday, 4/27
Speaker: Nemo (the Odyssey)

Tuesday, 5/4
Title: Shellable Dissections for Root Polytopes of Graphs and Digraphs
Speaker: Lilla Tóthmérész (Eötvös)
Time: 1:15-2:15

The root polytope of a bipartite graph $G=(U,W,E)$ is a polytope in $\mathbb{R}^{U\cup W}$ defined as Conv$\{\mathbf{1}_u -\mathbf{1}_w : u \in U, w \in W, uv \in E\}$. This is a well-studied polytope that has nice connections to graph theory. For example its dimension is $|U|+|W|-2$ and its maximal simplices correspond to spanning trees of the graph.

I examine the root polytope of directed graphs $G=(V,E)$, defined as Conv$\{\mathbf{1}_h -\mathbf{1}_t : \overrightarrow{th} \in E\}$. The root polytope of a bipartite (undirected) graph is a special case in which we orient each edge towards $U$.

The root polytope of a digraph might have dimension $|V|-1$. I am interested in the case that the dimension is $|V|-2$ (as in the undirected case). This turns out to be true if each cycle of the digraph has equal numbers of edges going in the two cyclic directions. We call these digraphs semibalanced.

A reason why root polytopes of semibalanced digraphs are interesting is that the facets of so-called symmetric edge polytopes (yet another class of polytopes associated to graphs, whose volume has a relevance to physics) are root polytopes of semibalanced digraphs.

I show a shellable dissection for root polytopes of semibalanced digraphs. Combinatorially, this means giving a nice set of spanning trees. The elements of the $h$-vector also have a combinatorial meaning in terms of the spanning trees.

I also tangentially touch the theory of greedoids.

This is joint work with Tamás Kálmán.

Zoom meeting https://binghamton.zoom.us/j/95302383985

Tuesday, 5/11
Title: Expressing the Skew Spectrum of an Oriented Graphs in Terms of the Spectrum of an Associated Signed Graph
Speaker: Zoran Stanić (Belgrade)
Time: 1:15-2:15

For an oriented graph G′ = (G,σ′) and a signed graph G ̇ = (G,σ), both underlined by the same finite simple graph G, we say that the signature σ is associated with the orientation σ′, and simultaneously that G ̇ is associated with G′, if σ(ik)σ(jk) = s_iks_jk holds for every pair of edges ik and jk, where (s_ij) is the skew adjacency matrix of G′. We prove that such a signature and orientation exist if and only if G is bipartite. On the basis of this result, we prove that, in the bipartite case, the skew spectrum of G′ is fully determined by the spectrum of an associated signed graph G ̇, and vice versa. In the non-bipartite case, we prove that the skew spectrum of G′ is fully determined by the spectrum of a signed graph associated with the bipartite double of G′. In this way, we show that the theory of skew spectra of oriented graphs has a strong relationship with the theory of spectra of signed graphs. In particular, a problem concerning the spectrum of an oriented graph can be transferred to the domain of signed graphs, considered there (where we deal with spectra of real symmetric matrices) and then the result can be ‘returned’ to the framework of oriented graphs. We apply this approach to some particular problems.

Zoom meeting https://binghamton.zoom.us/j/95302383985

Tuesday, 5/18
Title: The Varchenko-Gel’fand Ring of a Hyperplane Arrangement or a Cone
Speaker: Galen Dorpalen-Barry (Minnesota)
Time: 1:15-2:15

The coefficients of the characteristic polynomial of an arrangement in a real vector space have many interpretations. An interesting one is provided by the Varchenko-Gel’fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise multiplication. Varchenko and Gel’fand gave a simple presentation for this ring, along with a filtration whose associated graded ring has its Hilbert function given by the coefficients of the characteristic polynomial. I generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes.

Time permitting, I will discuss Varchenko–Gel’fand analogues of some well-known results in the Orlik–Solomon algebra regarding Koszulity and supersolvable arrangements.

Zoom meeting https://binghamton.zoom.us/j/95302383985

FALL 2020

Tuesday, 9/8
Title: Like speed dating, except combinatorics
Speakers: Michael Dobbins, Laura Anderson, Tom Zaslavsky
Time: 1:10-2:10