Fall 2024

Unless otherwise noted, all talks will take place in Thornton Hall 211 at San Francisco State University.

Wednesday, September 11, 4:00 - 5:00

Jonathan Niño-Cortes

University of Washington

Title: The convex algebraic geometry of higher-rank numerical ranges

Abstract: The higher-rank numerical range is a convex compact set generalizing the classical numerical range of square complex matrices, first appearing in the study of quantum error correction. In this talk, I will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn’s theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix.

Wednesday, September 18, 4:00 - 5:00

Michelle Wachs

University of Miami & SLMath

Title: Genocchi numbers and hyperplane arrangements

Abstract: In joint work with Alex Lazar, we refine a result of Gabor Hetyei relating the number of regions of a homogenized version of the Linial hyperplane arrangement to the median Genocchi numbers. We do so by obtaining combinatorial interpretations of the coefficients of the characteristic polynomial of the arrangement and by deriving generating functions for the characteristic polynomials, which reduce to known generating functions for the Genocchi and median Genocchi numbers. Our work involves the Ferrers graphs of Ehrenborg and van Willigenburg, a class of permutations related to Dumont permutations, the surjective staircase tableaux of Dumont, and a result of Chung and Graham on chromatic polynomials of incomparibility graphs. Our techniques also yield type B analogs, and Dowling arrangement generalizations.

Wednesday, September 25, 4:00 - 5:00

Theo McKenzie

Stanford University

Title: Constructing vertex expanding graphs

Abstract: In a vertex expanding graph, every small subset of vertices neighbors many different vertices. Random graphs are near-optimal vertex expanders; however, it has proven difficult to create families of deterministic near-optimal vertex expanders, as the connection between vertex and spectral expansion is limited. We discuss successful attempts to create unique neighbor expanders (a weak version of vertex expansion), as well as limitations in using common combinatorial methods to create stronger expanders. This is based on joint work with Jun-Ting Hsieh, Sidhanth Mohanty, and Pedro Paredes.