I will describe a new proof, joint with Adam Piggott (UQ), that groups presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 are exactly the plain groups (free products of finite and infinite cyclic groups). Our proof relies on a new result about properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.
A sequence of expanders is a family of finite graphs that are sparse yet highly connected. Such families of graphs are fundamental object that found a wealth of applications throughout mathematics and computer science. This talk is centred around an 'asymptotic' weakening of the notion of expansion. The original motivation for this asymptotic notion comes from the study of operator algebras associated with metric spaces. Further motivation comes from some recent works which established a connection between asymptotic expansion and strongly ergodic actions. I will give a non-technical introduction to this topic, highlighting the relations with usual expanders and group actions.
This is joint work with Alicia Kollár.
This talk is related to this arXiv paper.
This video is of the London Mathematical Society‘s 2020 Hardy Lecture, held on Zoom.
In their seminal paper Erdős and Szemerédi formulated conjectures on the size of sum set and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph, when we consider sums and products of some pairs only. With Noga Alon and Imre Ruzsa we showed that this strong form of the Erdős-Szemerédi conjecture does not hold. In this talk I will list some related problems and recent results.
We improve the upper bound for diagonal Ramsey numbers to R(k+1,k+1) ≤ exp(-c(log k)2)(2k)!/(k!)2 for k ≥ 3. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal 'effective quasirandomness' results about convergence of graphs. This optimality represents a natural barrier to improvement.
We are interested in Laplacians on graphs associated with finitely generated groups: Cayley graphs and, more generally, Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general. We will discuss various techniques that allow to construct examples with different types of spectra - connected, union of two intervals, totally disconnected - and with various types of spectral measure. The problem of spectral rigidity will also be addressed.
The intersection of the division group of a finitely generated subgroup of a torus with an algebraic sub-variety has been understood for some time (Lang, Laurent). After a brief review of some of the tools in the analysis and their recent extensions (André-Oort conjectures), we give some old and new applications; periodicity of Betti numbers, algebraicity of Painlevé equations, and the additive structure of the spectra of quantum graphs.
Ribbon graphs capture the topology of open Riemann surfaces in an elementary combinatorial form. One can hope this is the first step toward a general theory for open symplectic manifolds such as Stein manifolds. We will discuss progress toward such a higher-dimensional theory (joint work with Alvarez-Gavela, Eliashberg, and Starkston), and in particular, what kind of topological spaces might generalize graphs. We will also discuss applications to the calculation of symplectic invariants.
Given a finite group G and a set A of generators, the diameter diam(Γ(G,A)) of the Cayley graph Γ(G,A) is the smallest 𝓁 such that every element of G can be expressed as a word of length at most 𝓁 in A ⋃ A-1. We are concerned with bounding diam(G):= maxA diam(Γ(G,A)). It has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n. In 2011, Helfgott and Seress gave a quasipolynomial bound exp((log n)4+ε). We will discuss a recent, much simplified version of the proof.
We present an approach to show the existence of large expanders in locally sparse graphs and in sparse (including super-critical) random graphs, as well as its consequences for extremal questions and positional games.
