Name: 2024 09 08 Seminar
Start: 2024-09-16
End: 2024-12-13

Seminar Autumn 2024

The seminars take place on Wednesdays from 2-3 pm during term time in Horten Lecture Theatre 1 (HLT 1). They are aimed to appeal to most of our staff and PhD students. They are followed by coffee, tea and biscuits in the McCrea building (next to room 0-08). Most times we will go to dinner afterwards.

Everyone is welcome!

18th September: Jan-Christoph Schlage-Puchta (University of Rostock)

Title: Numerical computation of the Witten zeta function of symmetric groups and Fenchel's conjecture

Let G be a group. Then define ζ_G(s), the Witten ζ-function of G, as the sum over all χ in Irr(G) of χ(1)^-s, where Irr(G) denotes the set of irreducible characters. The series of functions ζ_Sn(s) has first been studied by Lulov in the connection with random walks on S_n, later it was used to study the subgroup growth of surface groups and Fuchsian groups.

While the behaviour of ζ_Sn(s) is easy to describe for when n tends to infinity, computing values for specific values of n and s is computationally demanding, as the number of representations of S_n grows like exp(π √(2n/3)).

Here we give an algorithm that produces reasonable approximations for ζ_Sn(s) for medium sized values of n. As applications we prove a stronger versions of Fenchel's conjecture: Every Fuchsian group contains a torsion free subgroup of the minimal index for which such a subgroup is not prohibited by congruence restrictions.

2nd October at 3pm: Robert Johnson (Queen Mary, University of London)

Title: Correlation and Intersection: from Sets to Permutations via Orders

Starts at 3pm!

We will discuss two extremal topics on permutations.

The first is analogues of the Harris-Kleitman inequality for families of permutations. It turns out that there are two natural notions of what it means for a family of permutations to be an up-set (corresponding to the strong and weak Bruhat orders) and surprisingly the correlation that occurs in the two cases is quite different. The second is a new notion of intersection for permutations.

Through these two examples, we will see a framework for translating extremal set theory concepts into the realm of permutations which has the potential to yield many open questions.

This is joint work with Imre Leader and Eoin Long.

9th October: Heather Leitch (Royal Holloway, University of London)

16th October: Antony Hilton (University of Reading)

Title: The total chromatic number of a graph

23rd October: Will Cohen (University of Cambridge)

30th October: Álvaro Gutiérrez Cáceres (University of Bristol)

6th November: Steve Lester (King's College London)

13th November: Olivea Reade (Open University)

11th December: Julia Wolf (University of Cambridge)