University of Bristol, Wednesday February 7th 2018

Venue: Lecture theatre SM1, School of Mathematics, University of Bristol

**Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine – note that most people in this category are elgibile for a rail-card).**

14:00-15:00 Carlisle King (Imperial College, London) Probabilistic methods and generation of finite simple groups

Let *G* be a finite group. It is well-known that *G* is generated by a pair of elements – we say *G* is 2-generated. Later, it was shown that 2 elements chosen at random from *G* generate *G* with probability tending to 1 as |*G*| tends to infinity. A natural refinement is to ask, given a pair of integers (*a,b*), whether *G* is 2-generated by an element of order *a* and an element of order *b*. If such a pair exists, we say that *G* is (*a,b*)-generated. We will explore some past results regarding (2,3)-generation proved using probabilistic methods, as well as a recent result on (2,*p*)-generation for some prime *p*.

An equivalent statement of the 2-generation theorem is that every finite simple is an image of *F _{2}*, the free group on 2 generators. More generally, given a finitely presented group Γ, one can ask which finite simple groups are images of Γ. We will study a new result in this area concerning the free produc Γ=

*A**

*B*of nontrivial finite groups

*A*and

*B*, also proved using probability methods.

15:00-15:30 Break

15:30-16:30 Stacey Law (Cambridge) *p’*-branching for symmetric groups

Let *p* be any prime and *n* be any natural number. Let χ be an ordinary irreducible character of the symmetric group *S _{n}* whose degree is coprime to

*p*.

We bound the number of *p’*-irreducible constituents of the restriction of χ to *S _{n−1}*. This generalizes work of Ayyer, Prasad and Spallone (2016) for the prime

*p = 2*. This is joint work with E. Giannelli and S. Martin.

16:30-17:30 Nick Gill (South Wales) Cherlin’s conjecture, Lachlan’s theory of homogeneous relational structures and the notion of “sporadicness”

Some years ago, Lachlan advanced a theory on homogeneity in relational structures which imposed a natural “hierarchy of complexity” on the universe of homogeneous relational structure. This theory was reworked by Cherlin in the 1990’s with a view to understanding finite permutation groups from a model theoretic point of view.

One upshot of all this is that we know the existence of an infinite family of theorems describing the so-called “relational complexity” of all finite permutation groups. The problem is that, although we know these theorems exist, and even have a “form” for them, nonetheless we do not yet have the precise statement of any of them. However Cherlin has conjectured what (part of) the first of these theorems should say, and we will discuss this conjecture at some length.

There has also been substantial progress on this conjecture due to Cherlin himself, to Wiscons, and to myself and various co-authors. In particular the recent results that I will describe are due to myself, Pablo Spiga, Francis Hunt and Francesca Dalla Volta.

The talk has elements of model theory, combinatorics and finite permutation groups, and should be accessible to all.