All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 2 pm, unless stated otherwise. Tea will be served after the seminar at 3pm in Room 237 of the McCrea Building. All are welcome!

**Seminars for the spring term 2011.**

- 11 January: Mathematics group meeting
- 18th January: Cecelia Busuioc (RHUL): Eisenstein congruences and K-theory (abstract)
- 25th January: Simon Goodwin (Brimingham): Representations of finite W-algebras (abstract)
- 1st February: Richard Mycroft (QMUL): Perfect matchings and packings in graphs and hypergraphs
- 8th Februrary: John Britnell (Bristol): Coverings of finite linear groups by proper subgroups
- 15th February: Kevin Buzzard (Imperial College): Artin's conjecture on L-functions (abstract)
- 22th February: Mark Damerell (RHUL): The walrus & the carpenter, an example of the way that obscure bits of mathematics appear in unlikely places.
- 1st March: Heidi Gebauer (ETH Zuerich): Game theoretic Ramsey numbers (abstract)
- 8th March: Brita Nucinkis (Southampton): Cohomological finiteness properties of the Brin-Thomspon-Higman groups sV (abstract)
- 15th March: Rowena Paget (Kent): Set partitions and symmetric groups
- 22th March: Oscar Marmon: Sums and differences of four k-th powers (abstract)

**Abstracts:**

Cecilia Busuiok: Einstein congruences and K-theory

In this talk, we will investigate congruences of periods of parabolic modular forms at Eisenstein primes and their connection to the arithmetic of the K2-groups of rings of integers. Our approach to this study deals with an explicit construction of Eisenstein

cohomology classes in the parabolic cohomology of the arithmetic group ¡0(N).

Simon Goodwin: Representations of finite W-algebra

Finite W-algebras are certain associative algebras that can be viewed as the enveloping algebra of the Slodowy slice to a nilpotent orbit in a reductive Lie algebra g. The representation theory of W-algebras has a number of important applications: in particular, to the study of the primitive ideals of the universal enveloping algebra of g.

In this talk, I will give an overview of the representation theory of finite W-algebras. All terms above will be explained and motivated.

Kevin Buzzard: Artin's conjecture on L-functions

I will tell the story of what little we know about an old conjecture of Emil Artin from the 1930s. The conjecture is deceptively simple. The Riemann zeta function is defined by a series in s which only converges for Re(s)>1, but the function is well-known to have a meromorphic continuation to the complex plane and to satisfy a functional equation. Artin constructed some more general types of zeta function, defined by series which converged for Re(s)>1, and conjectured that they also should have a meromorphic continuation. This simple-sounding conjecture is still wide wide open and recent incremental progress in our knowledge about it has only come via recent breakthroughs in subjects like the theory of modular forms. I will explain what little we know. I will not assume prior knowledge of Galois representations or modular forms or anything like this and there will be no proofs presented! It will be more of a survey of the area.

Heidi Gebauer: Game theoretic Ramsey numbers

The Ramsey Number, R(k), is defined as the minimum N such that every 2-coloring of the edges of K_{N} (the complete graph on N vertices) yields a monochromatic k-clique. For 60 years it is known that 2^(k/2) < R(k) < 4^k, and it is a widely open problem to find significantly better bounds. In this talk we consider a game theoretic variant of the Ramsey Numbers: Two players, called Maker and Breaker, alternately claim an edge of K_{N}. Maker's goal is to completely occupy a K_{k }and Breaker's goal is to prevent this. The game theoretic Ramsey Number R'(k) is defined as the minimum N such that Maker has a strategy to build a K_{k} in the game on K_{N}. In contrast to the ordinary Ramsey Numbers, R'(k) has been determined precisely -- a result of Beck. We will sketch a new, weaker result about R'(k) and use it to solve some related open problems.

Brita Nucinkis: Cohomological finiteness properties of the Brin-Thomspon-Higman groups sV

This is joint work with D. Kochloukova and C. Martinez-Perez.

We show that Brin's generalisations 2V and 3V of the Thompson-Higman group V are of type FP_\infty. Our methods also give a new proof that both groups are finitely presented.

Our proof is based on an idea of Ken Brown: In 1985 he showed that Thompson's groups F, T and V as well as some generalisations such as Higman's groups V_{n,r} are of type FP_\infty and are finitely presented. These groups are viewed as groups of algebra-automorphisms and act combinatorially on the geometric realisation of a poset determined by the algebra. It is then shown that this complex has a filtration yielding the required finiteness properties.

In this talk I will begin by explaining various ways to define the original Thompson groups, especially focusing on their description as certain automorphism groups of tree diagrams, and then move on to explaining how to generalise this to sV. This will lead to the definition of the poset mentioned above.

Oscar Marmon: Sums and differences of four k-th powers

I will discuss the following problem: in how many different ways can we write a natural number N as the sum of k-th powers of integers? After a brief survey of relevant conjectures and results I will present some results of my own, giving new upper bounds for the number of representations of an integer as the sum or difference of four k-th powers. I will briefly outline the main ideas in the proof, which uses a version of the Bombieri-Pila determinant method and is a mix of analytic number theory and algebraic geometry.

**Useful links**:

Campus map (The McCrea Building is number 17)