Group theory – Page 25 – MathVideos.org

Harald Helfgott: The Diameter of the Symmetric Group: Ideas and Tools

3rd February, 2017

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):= max_A 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.

Doron Puder: Word-Measures on Unitary Groups

2nd February, 2017

One approach to studying properties of random walks on groups with random generators is to study word-measures on these groups. This approach was proven useful for the study of symmetric groups and random regular graphs. In the current work we focus on the unitary groups U(n). For example, if w is a word in F₂ = <x,y>, sample at random two elements from U(n), A for x and B for y, and evaluate w(A,B). The measure of this random element is called the w measure on U(n). We study the expected trace (and other invariants) of a random unitary matrix sampled from U(n) according to the w-measure, and find surprising algebraic properties of w that determine these quantities.

Alireza Salehi Golsefidy: Super-approximation

31st January, 2017

I will report on the latest results on super-approximation. Roughly super-approximation gives us the right condition in order to get a family of expanders out of the Cayley graphs of the congruence quotients of a group generated by finitely many rational matrices. I will mention a sum-product phenomenon in number fields which is used in the proof of super-approximation. Some of the applications of super-approximation will be mentioned at the end.

Tag - Group theory