Fall 2022
December 2
Speaker: Stephen Stern
Title: An Introduction to Edge and Cover Ideals
Abstract: I will introduce two bijections between simple hypergraphs and square-free monomial ideals that are central in Combinatorial Commutative Algebra. These bijections allow us to exchange combinatorial problems for algebraic problems and vice versa. For example, I will show that the problem of finding the chromatic number of a (hyper) graph is equivalent to a certain ideal membership problem. The talk will be accessible to audience members with a basic understanding of ideals and graph theory, and I will include illustrative examples throughout.
November 18
Speaker: Josh Males
Title: Modular forms: why should we care?
Abstract: In this short talk I'll try to give a very brief introduction to a central theme in my research: modular forms. These are functions in the upper half-plane whose infinite symmetries force extraordinary properties. I'll keep the talk very light and accessible, and talk a little bit about the kinds of problems that I think a lot about at the intersection of number theory, and combinatorics.
Modular_Forms__Why_should_we_care_.pdf2022-11-18 12-40-02.mp4November 4
Speaker: Hermie Monterde
Title: An Introduction to Continuous-time Quantum Walks
Abstract: Continuous-time quantum walks (CTQW) were devised to understand how quantum information travels through a quantum spin network. Motivated by its applications to quantum computing, the study of CTQW has recently exploded in the last 20 years. In this talk, we will introduce the idea of a continuous-time quantum walk and some important types of quantum transmission that occur within a CTQW.
2022-11-04 12-32-37.mp4An Introduction to Continuous-time Quantum Walks.pdfOctober 21
Speaker: Ian Thompson
Title: Minimal Boundaries for Operator Algebras.
Abstract: We introduce boundaries for operator algebras, which are collections of irreducible *-representations that completely capture the spatial norm attainment for the algebra. In the classical setting of continuous functions, there is a minimal boundary for an algebra A in C(X) called the Choquet boundary, and a result of Bishop states that the Choquet boundary is the collection of peak points for the algebra A. We investigate the question of minimal boundaries for arbitrary operator algebras, and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. This is joint work with Raphaël Clouâtre.
October 7
Speaker: Tommy Cai
Title: The products of conjugates of elements in free groups.
Abstract: I will talk about two things. 1. A conjecture about the lengths of the product of conjugates of an element in free groups. 2. A proof that some cyclic subgroups of free groups are strong malnormal. (A subgroup C of a group A is strong malnormal if for any nontrivial element c in C and a_1,\dotsc, a_n in A, the product of conjugates a_1^{-1}ca_1\dotsm a_n^{-1}ca_n is always not in C if at least one a_i is not in C.)
These are two problems that appear in my research for my Ph.D. program.
2022-10-07 12-36-00.mp4