Events for 04/28/2023 from all calendars
Mathematical Physics and Harmonic Analysis Seminar
Time: 1:50PM - 2:50PM
Location: BLOC 302
Speaker: Lior Alon, MIT
Title: Quasicrystals and Lee-Yang Polynomials
Abstract: The concept of quasi-periodic sets, functions, and measures is prevalent in diverse mathematical fields such as Mathematical Physics, Fourier Analysis, and Number Theory. In natural science, Shechtman was awarded the 2011 Nobel Prize for the discovery of materials with quasi-periodic atomic structures, which are now known as Quasicrystals.
This talk will focus on Fourier quasicrystals (FQ): discrete measures with Fourier transform which is also discrete, and with some growth bound. In particular, we care about sets with a counting measure which is an FQ. By the Poisson summation formula, the counting measure of any discrete periodic set is an FQ. Recently, Kurasov and Sarnak provided a general construction (motivated by quantum graphs) of counting measures that are FQ. Their method is based on restricting the zero set of a multivariate Lee-Yang polynomial to an irrational line in the torus. In particular, they answered a long-standing question of Meyer, providing explicit FQ which is the counting measure of an a-periodic uniformly discrete set.
In this talk, we will show that the Kurasov-Sarnak construction generates all FQ counting measures and that generically these sets are a-periodic and uniformly discrete. If time permits, we will see that these measures have well-defined gaps distribution whose properties are deduced from the polynomial's structure. The talk is aimed at a broad audience, no prior knowledge in the field is assumed.
Based on joint works with Alex Cohen and Cynthia Vinzant.
Algebra and Combinatorics Seminar
Time: 3:00PM - 4:00PM
Location: BLOC 302
Speaker: Patricia Klein, Texas A&M University
Title: Geometric vertex decomposition and liaison
Abstract: Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, we will describe an explicit connection between these approaches. In particular, we will describe how each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras. This connection also gives us a framework for implementing with relative ease Gorla, Migliore, and Nagel’s strategy of using liaison to establish Gröbner bases. Time permitting, we will describe briefly, as an application of this work, a proof of a conjecture of Hamaker, Pechenik, and Weigandt on diagonal Gröbner bases of certain Schubert determinantal ideals. This talk is based on joint work with Jenna Rajchgot.
Free Probability and Operators
Time: 4:00PM - 5:00PM
Location: BLOC 306
Speaker: Michael Anshelevich, TAMU
Title: Types of noncommutative independence and asymptotics of random matrices.
Abstract: We will discuss how the asymptotic joint distribution of two random matrices "in a general position" can be described using cyclic c-freeness and its particular cases; or, conversely, how different types of noncommutative independence can be asymptotically modeled by random matrices. In the first talk we will concentrate on algebraic independence theories; in the second and third talks, we will explain the connection with random matrices. The talks are based primarily on the articles arXiv:2207.06249 by Cébron and Gilliers; and arXiv:2205.01926 by Cébron, Dahlqvist, and Gabriel.
Geometry Seminar
Time: 4:00PM - 5:00PM
Location: BLOC 302
Speaker: Derek Wu, TAMU
Title: Border rank bounds for GL_n-invariant tensors arising from spaces of matrices of constant rank
Abstract: One measure of the complexity of a tensor is its border rank. Finding the border rank of a tensor, or even bounding it, is a difficult problem that is currently an area of active research, as several problems in theoretical computer science come down to determining the border ranks of certain tensors. For a class of $GL(V)$-invariant tensors lying in a $GL(V)$-invariant space $V\otimes U\otimes W$, where $U$ and $W$ are $GL(V)$-modules, we can take advantage of $GL(V)$-invariance to find border rank bounds for these tensors. I discuss a special case where these tensors correspond to spaces of matrices of constant rank.