Events for 11/11/2022 from all calendars

Student/Postdoc Working Geometry Seminar

Time: 1:00PM - 2:15PM

Location: BLOC 628

Speaker: JM Landsberg, TAMU

Title: A problem in graph theory and algebraic geometry

Abstract: The problem: let K_s,t denote the complete bipartite graph with s edges on the left and t on the right. What is the largest number of edges in an n vertex graph not containing K_s,t as a subgraph? The use of algebraic geometry: a suitable random subvariety on the Segre P^b\times P^b over F_q furnishes the vertex set and the zero set of a random polynomial on the Segre furnishes the edges. This is work of Boris Bukh. In his proof he has to avoid a bad set which has interesting geometry that I'll discuss.

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 306

Speaker: Gamal Mograby, Tufts University

Title: Topological quantum numbers

Abstract: We present a detailed spectral analysis for a new class of fractal-type diamond graphs and provide a gap-labeling theorem in the sense of Bellissard for the corresponding probabilistic graph Laplacians using the technique of spectral decimation. Labeling the gaps in the Cantor set by the integrated density of states provides a set of topological quantum numbers that reflect the branching parameter of the graph construction and the decimation structure. The spectrum of the natural Laplacian on limit graphs is shown generically to be pure point supported on a Cantor set. However, one particular graph has a mixture of pure point and singularly continuous components.

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

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).

Free Probability and Operators

Time: 4:00PM - 5:00PM

Location: BLOC306, ONLINE

Speaker: Jacob Campbell, University of Waterloo

Title: Commutators in finite free probability

Abstract: I will present a recent preprint (arXiv:2209.00523) concerning the commutator in finite free probability. The main result is an explicit formula for the expected characteristic polynomial of AUBU^* - UBU^*A, where A and B are fixed dxd matrices and U is a dxd random unitary matrix, in terms of the respective characteristic polynomials of A and B. The main ideas are to use Weingarten calculus to reduce the problem to one of combinatorial representation theory, and then use a 1992 result of Goulden and Jackson to compute immanants of certain low-rank matrices. Time permitting, I will suggest some potential avenues for finding the d-finite version of the Nica-Speicher formula for the free cumulants of commutators of free variables. ZOOM LINK: https://tamu.zoom.us/j/99940122674