Mathematics Colloquia, Fall 2021

October 21, 2021

Time: 4 p.m. CST 
Speaker: Thomas Schlumprecht, Texas A &M
Title:  Lamplighter metric spaces and their embeddings into $L_1$
Abstract: Understanding how a group or a graph, viewed as a geometric object, can be faithfully embedded into certain Banach spaces is a fundamental topic with applications to geometric group theory and theoretical computer science.

In this joint work with Florent Baudier, Pavlos Motakis and Andras Zsak we observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the $L_1$-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a $n$-point metric space embeds bi-Lipschitzly into $L_1$ with distortion $O(\log n)$. 

In particular, for every finite group $G$ the lamplighter group $H = \mathbb{Z}_2\wr G$ bi-Lipschitzly embeds into $L_1$ with distortion $O(\log\log|H|)$.

In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into $L_1$ of the lamplighter group over the $d$-dimensional infinite lattice $\mathbb Z^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite $d$-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

Via Zoom: Please contact denka@illinois.edu or currid@illinois.edu for link. The Zoom link will activate at 3:45 p.m.

October 7, 2021

Time: 4 p.m. CST 
Speaker:  John Shareshian, Washington University of St. Louis
Location: 245 Altgeld Hall
Title: A problem on divisors of binomial coefficients, and a theorem on noncontractibility of coset posets
Abstract: Fix an integer n>1. It follows directly from a theorem of Kummer that the greatest common divisor of the members of the set BC(n) nontrivial binomial coefficients nC1,nC2,…nC(n-1) is one unless n is a prime power. With this in mind, we define b(n) to be the smallest size of a set P of primes such that every member of BC(n) is divisible by at least one member of P. In joint work with Russ Woodroofe, we ask whether b(n) is at most two for every n. The question remains open.

I will discuss what we know about this question, and how we discovered it during our investigation of a problem raised by Ken Brown about certain topological spaces: Given a finite group G, let C(G) the set of all cosets of all proper subgroups of a finite group, partially ordered by containment. The order complex of C(G) is the simplicial complex whose k-dimensional faces are chains of size k+1 from C(G). We show that this order complex has nontrivial reduced homology in characteristic two, and is therefore not contractible.

If time permits, I will discuss also related work on invariable generation of simple groups, joint with Bob Guralnick and Russ Woodroofe.

September 30, 2021

Time: 4 p.m. CST 
Speaker: Bernard Lidicky, Iowa State University
Title:  Flag algebras and its applications
Abstract: Flag algebras is a method, developed by Razborov, to attack problems in extremal combinatorics. Razborov formulated the method in a very general way which made it applicable to various settings. The method was introduced in 2007 and since then its applications have led to solutions or significant  improvements of best bounds on many long-standing open problems, including problems of Erd\H{o}s.The main contribution of the method was transferring problems from finite settings to limits settings. This  allows for clean calculations ignoring lower order terms. The method can utilize semidefinite programming  and computers to produce asymptotic results. This is often followed by stability type arguments with the  goal of obtaining exact results.

In this talk we will give a brief introduction of the basic notions used in flag algebras and demonstrate  how the method works. Then we will discuss applications of the flag algebras in different settings.

Time: 4 p.m. CST 
Speaker: Thomas Shlumprecht, Texas A & M University
Location: TBD
Title:  To come
Abstract: To come