Pure Mathematics Seminar

Fridays 3:15 - 4:15pm

Peter Hall Building, Room 162

26 July:

Student Pure Mathematics Seminar: David Lumsden

Card shuffling, the symmetric group and spectral gaps

In this talk I will give an overview of some results related to shuffling cards viewed as a random walk on the symmetric group. In particular the Aldous spectral gap conjecture and results surrounding generating a permutation from random transpositions will be discussed.

2 August:

Pure Mathematics Seminar: Yuxuan Li

Aldous' spectral gap conjecture and its generalizations

Aldous' spectral gap conjecture asserts that for any finite graph $\Gamma$ with vertex set $[n]=\{1, 2, \ldots, n\}$, the interchange process and the random walk on $\Gamma$, both continuous-time Markov chains, exhibit identical spectral gaps. Notably, the random walk with the state space $[n]$ is a subprocess of the interchange process, which operates over the larger state space $S_n$. After nearly two decades of being unresolved, this conjecture was conclusively validated in 2010.

This presentation aims to introduce several equivalent formulations of Aldous' conjecture from diverse perspectives. Additionally, it will explore various generalizations of the conjecture, emphasizing insights drawn from algebraic graph theory.

9 August:

Student Pure Mathematics Seminar

16 August:

Pure Mathematics Seminar: Jonathan Bowden

The tight geography problem for high dimensional contact manifolds

Contact manifolds arise naturally in the context of classical mechanics as regular level sets of Hamiltonians in phase space satisfying certain convexity properties. More precisely, a contact structure is a totally non-integrable hyperplane field, with classical examples given by left invariant fields on certain Lie groups. Relatively recently  Borman-Elishberg-Murphy proved an h-principle for so-called overtwisted contact structures, reducing the existence and classification problem to classical obstruction theory. This leads to the problem of studying contact manifolds that exhibit rigidity, so-called tight contact structures. In this talk I will report on some progress on a programme to utilise classical surgery theory together with analytic tools such as Floer homology to construct examples of such tight contact structures. (This is part of joint work with D. Crowley and J. Hammet)