This page lists this term's Pure Mathematics Seminars. All are welcome. All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 2.00pm, unless stated otherwise. Tea is served after the seminar at 3.00 in Room 237 of the McCrea Building.

**Seminars for the autumn term 2008.**

- 30th September:
**Hugh Montgomery**(University of Michigan, Ann Arbor), Analytic inequalities. - 7th October:
**David Craven**(Oxford), Symmetric groups, partitions and representation growth (abstract). - 14th October:
**Mark Jerrum**(QMUL), An approximation trichotomy for #CSP. - 21st October:
**Titus Hilberdink**(Reading), Well behaved generalised primes and integers - 28th October:
**Malwina Luczak**(LSE), Glauber dynamics for the Mean-field Ising Model: cut-off, critical power law, and metastability (abstract). - 4th November:
**Henri Johnsten**(Oxford), Non-existence and splitting theorems for normal integral bases. - 11th November:
**James McKee**(RHUL), Galois theory of Salem polynomials (abstract) - 18th November:
**Nigel Boston**(University College Dublin), Random groups in number theory and topology (abstract) - 25th November:
**Graham Everest**(East Anglia), Elliptic Divisibility Sequences and Hilbert's Tenth Problem - 2nd December:
**Tim Riley**(Bristol) Dual spanning trees in planar graphs, and "filling invariants" for finitely presented groups (abstract)

**Abstracts:**

David Craven: Symmetric groups partitions, and representation growth.

The representation growth of residually finite groups -- studying the dimensions of irreducible complex representations of a, usually infinite, group -- is a relatively new area. Of particular interest are the degrees of characters of symmetric groups, which are determined by combinatorial objects associated with partitions. In this talk I will discuss the multiplicities of character degrees of symmetric groups, and an application to the field representation growth.

Malwina Luczak: Glauber dynamics for the Mean-field Ising Model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For \beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-\beta)]^{-1} n\log n. For \beta = 1, we prove that the mixing time is of order n^{3/2}. For \beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n \log n). This is joint work with David Levin and Yuval Peres.

James McKee: Galois theory of Salem polynomials

This is joint work with Christos Christopoulos. Let f(x) be the minimal polynomial of a Salem number, and let g(x) be the associated "trace polynomial". Let F and G be the Galois groups of f(x) and g(x). The group G is a quotient of F by a normal subgroup N. The group extension splits, so that F is isomorphic to the semidirect product of N and G.

Nigel Boston, Random groups in number theory and topology

In a recent Inventiones paper Dunfield and Thurston compare fundamental groups of random 3-mainfolds and random discrete groups. Analogously, we develop a theory comparing certain random Galois

groups and random p-groups. In particular, what is the probability that r relators chosen at random from a g-generator free group present a particular group? We make sense of and answer this question for p-groups. This leads to some mysterious mass formulae, some infinite groups that arise with nonzero probability, and applications to number theory.

Tim Riley: Dual spanning trees in planar graphs, and "filling invariants" for finitely presented groups

I will describe some work with W.P.Thurston on how duality conditions affect the existence of efficient spanning trees in planar graphs. I will explain the significance of the topic for the study of filling invariants for finitely presented groups.

**Useful links**:

Campus map (The McCrea Building is number 17)