| Date: | August 26, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Galen Dorpalen-Barry, Ruhr-Universität Bochum |
| Title: | Real Hyperplane Arrangements and the Varchenko-Gelfand Ring |
| Abstract: | For a real hyperplane arrangement A, Varchenko--Gelfand ring is the ring
of functions from the chambers of A to the integers with pointwise
addition and multiplication. Varchenko and Gelfand 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
Poincaré polynomial. Their work was extended to oriented matroids by
Gelfand—Rybnikov, who gave an analogous presentation and filtration.
We extend this work first to pairs (A,K) consisting of an arrangement A
in a real vector space and open convex set K, and then to conditional
oriented matroids. Time permitting, we will discuss an interesting
special case arising in Coxeter theory: Weyl cones of Shi arrangements.
We find that the coefficients of the cone Poincaré polynomial of a Weyl
cone are described by antichains in the root poset.
This talk contains joint work with Christian Stump, Nicholas Proudfoot,
and Jayden Wang. |
|
| Date: | September 16, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Youngho Yoo, TAMU |
| Title: | A unified Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups |
| Abstract: | Erdős and Pósa showed in 1965 that cycles obey an approximate packing-covering duality. While odd cycles do not satisfy such a duality, Reed proved that the only obstruction is the presence of a certain projective planar grid. In this talk we discuss generalizations of these results. Namely, in undirected group-labelled graphs, we characterize the topological obstructions to the Erdős-Pósa property of cycles with "allowable" group values, under some additional assumptions on the structure of the set of allowable values. This recovers many known results in the area and resolves a question of Dejter and Neumann-Lara from 1987 on characterizing when cycles of length L mod M satisfy the Erdős-Pósa property. Joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum. |
|
| Date: | October 7, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Matthew Faust, TAMU |
| Title: | On the Irreducibility of Bloch and Fermi Varieties |
| Abstract: | Given an infinite ZZ^n periodic graph G, the Schrodinger operator acting on G is a graph Laplacian perturbed by a potential at every vertex. Complexifying and choosing an M-periodic potential for some full rank free module M of ZZ^n fixes a representation of our operator as a finite matrix whose entries are Laurent polynomials. The vanishing set of the characteristic polynomial yields the Bloch variety, the vanishing set for fixed eigenvalues gives the Fermi variety. Questions regarding the algebraic properties of these objects are of significant interest in mathematical physics. We will focus our attention on the irreducibility of these varieties. Understanding the irreducibility of Bloch and Fermi varieties is important in the study of the spectrum of periodic operators, providing insight into the structure of spectral edges, embedded eigenvalues, and other applications. In this talk we will present several new criteria for obtaining irreducibility of Bloch and Fermi varieties for infinite families of discrete periodic operators. This is joint work with Jordy Lopez. |
|
| Date: | October 21, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Catherine Yan, TAMU |
| Title: | On the Limiting Vacillating Tableaux for Integer Sequences |
| Abstract: | A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence to a pair consisting of a standard Young tableau and a vacillating tableau. In this talk we show that for a given integer sequence $i$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(i)$ become stable when n goe to infinite; the limit is called the limiting vacillating tableau for $i$. We give a characterization of the set of limiting vacillating tableaux and present explicit formulas that enumerate those vacillating tableaux. This is a joint work with Zhanar Berikkyzy, Pamela Harris, Anna Pun and Chenchen Zhao. |
|
| Date: | October 28, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Daoji Huang, U of Minnesota |
| Title: | Bumpess pipe dream RSK, growth diagrams, and Schubert structure constants |
| Abstract: | The cohomology ring of the complete flag variety has a basis given by classes of the Schubert varieties. A central open question in Schubert calculus is to give a combinatorial interpretation of the multiplicative structural constants of the Schubert classes. While the general question remains open, in the Grassmannian case, the Schubert structure constants are known as Littlewood-Richardson coefficients and well-understood, and many of these classical rules are based on tableaux combinatorics. In this talk, we aim to generalize some of these results using bumpless pipe dreams. In particular, we introduce analogs of left and right RSK insertion for Schubert calculus of complete flag varieties. The objects being inserted are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. As an application, we adopt Lenart's growth diagrams of permutations to give a combinatorial rule for Schubert structure constants in the separated descent case. |
|
| Date: | November 4, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Christopher O'Neill , San Diego State University |
| Title: | Numerical semigroups, minimal presentations, and posets |
| Abstract: | A numerical semigroup is a subset S of the natural numbers that is closed under addition. One of the primary attributes of interest in commutative algebra are the relations (or trades) between the generators of S; any particular choice of minimal trades is called a minimal presentation of S (this is equivalent to choosing a minimal binomial generating set for the defining toric ideal of S). In this talk, we present a method of constructing a minimal presentation of S from a portion of its divisibility poset. Time permitting, we will explore connections to polyhedral geometry.
No familiarity with numerical semigroups or toric ideals will be assumed for this talk. |
|
| Date: | November 11, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Oliver Pechenik, U of Waterloo |
| Title: | Geometry of quasisymmetric functions |
| Abstract: | The combinatorics of symmetric function theory plays a central role both in combinatorial representation theory (of symmetric and general linear groups) and in enumerative geometry (through the cohomology of Grassmannians). The latter connection yields "K-analogues" of the classical symmetric function bases and their combinatorics by enriching the cohomology of Grassmannians to their K-theory rings. Quasisymmetric functions (QSym) are analogues of symmetric functions introduced by Stanley and Gessel in the 70s for primarily enumerative reasons, but also with a key role in the representation theory of 0-Hecke algebras. However, analogous connections to geometry and topology have been missing. In particular, although there has significant interest in "K-analogues" of quasisymmetric functions, there has been no known space whose K-theory they describe. We build on work of Baker & Richter (2008) to identify a loop space with a cellular cohomology basis corresponding to a classical basis of QSym. We then introduce an instance of "cellular K-theory," yielding the first geometrically-interpreted K-basis of QSym. Our polynomials are similar to ones introduced by Lam & Pylyavskyy (2007) and yet are new. This is joint work with Matt Satriano (arXiv:2205.12415). |
|
| Date: | November 18, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Frank Sottile, TAMU |
| Title: | A Murnaghan-Nakayama formula in quantum Schubert calculus |
| Abstract: | The Murnaghan-Nakayama formula expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. In geometry, a Murnaghan-Nakayama formula computes the intersection of Schubert cycles with tautological classes coming from the Chern character. In previous work with Morrison, we establshed a Murnaghan-Nakayama formula in the cohomology of a flag variety and conjectured a version for the quantum cohomology ring of the flag variety. In this talk, I will discuss some background, and then some recent work proving this conjecture. This is joint work with Benedetti, Bergeron, Colmenarejo, and Saliola. |
|
| Date: | November 30, 2022 |
| Time: | 2:30pm |
| Location: | BLOC 302 |
| Speaker: | Natasha Blitvic, Queen Mary University of London |
| Title: | Moment sequences in combinatorics |
| Abstract: | Take your favorite integer sequence. Is this sequence a sequence of moments of some probability measure on the real line? We will look at a number of interesting examples (some proven, others merely conjectured) of moment sequences in combinatorics. We will consider ways in which this positivity may be expected (or surprising!), the methods of proving it, and the consequences of having it. The problems we will consider will be very simple to formulate, but will take us up to the very edge of current knowledge in combinatorics, 'classical' probability, and noncommutative probability. |
|
| Date: | December 2, 2022 |
| Time: | 3:00pm |
| Location: | BLOC302 |
| Speaker: | Jan Draisma, Universitat Bern, Switzerland |
| Title: | A tensor restriction theorem over finite fields |
| Abstract: | The theorem in the title says that tensors of a fixed format
over a fixed finite field K are well-quasi-ordered by restriction: they
contain no infinite anti-chains. The same holds, more generally, for
"tensors" in spaces described by any finite-length functor from the
category of finite-dimensional K-vector spaces to itself.
I will discuss several equivalent versions and consequences of the
tensor restriction theorem, and explain what their proof reveals about
the coarse structure of arbitrary restriction-closed tensor properties.
I will also comment on analogous results for Zariski-closed tensor
properties over infinite fields, which were obtained earlier in
collaborations with Bik, Eggermont, and Snowden.
(Based on joint work with Andreas Blatter and Filip Rupniewski:
https://urldefense.com/v3/__https://arxiv.org/abs/2211.12319__;!!KwNVnqRv!CNJ4FaC_L_qPuDq9G0SSEEvvfBB16nXYFdC5foDxs-WtoqwWfZQ_EjR6RiKm7A_-Gq1IN2ghldDEZ2BAeVyhTD0$)
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