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 pm, unless stated otherwise. Tea will be served after the seminar at 3 pm in Room 237 of the McCrea Building.

**Seminars for the autumn term 2009.**

- 29th September:
**Tuvi Etzion**(Technion): Folding, Tiling, and Applications to Multidimensional Coding - 6th October:
**Imre Leader**(Cambridge): Higher Order Tournaments (abstract) - 13th October:
**Anton Evseev**(Cambridge): A refinement of the McKay conjecture (abstract) - 20th October:
**Tim Browning**(Bristol): The L1-norm of certain exponential sums (abstract) - 27th October:
**McDowell lecture.**No seminar - 3rd November:
**Daniel Appel**(RHUL): On the abelianization of standard congruence subgroups of the automorphism group of the rank two free group (abstract) - 10th November:
**Boris Bukh**(Cambridge): Sum-product theorems for polynomials (abstract) - 17th November:
**Alexander Stasinski**(Southampton): Deligne-Lusztig theory and generalisations (abstract) - 24th November:
**Vicky Neale**(Cambridge): Bracket quadratics as bases for the integers (abstract) - 1st December:
**Amaia Zugadi-Reizabal**Automorphisms of p-adic trees and Hausdorff dimension (abstract) - 8th December:
**Christian Elsholtz**(RHUL): Multidimensional problems in additive combinatorics

**Abstracts:**

Imre Leader, Higher order tournaments:

Given n points in general position in the plane, how many of the triangles formed by them can contain the origin? This problem was solved 25 years ago by Boros and Furedi, who used a beautiful translation of the problem to a non-geometric setting. The talk will start with background, including this result, and will then go on to consider what happens in higher dimensions in the geometric and non-geometric cases.

Anton Evseev: A refinement of the McKay conjecture

Let G be a finite group and N be the normalizer of a Sylow p-subgroup of G. The McKay conjecture, which has been open for more than 30 years, states that G and N have the same number of irreducible characters of degree not divisible by p (i.e. of p'-degree). The conjecture has been strengthened in a number of ways, in particular, by Alperin and Isaacs-Navarro. The latter refinement suggests a precise correspondence between irreducible character degrees of G and of N modulo p and up to sign, if one considers only characters of p'-degree. The talk will review some of these generalisations and will consider a possible new refinement, which implies the Isaacs- Navarro conjecture.

Tim Browning, The L1-norm of certain exponential sums

I will discuss the approach of Balog and Ruzsa for bounding below the L1-norm of linear exponential sums whose coefficients are supported on the square-free integers. I will discuss how their lower bound can be improved by linking a problem about spacing of fractions a/p^{2} to a problem about counting points of bounded height on elliptic curves. This is joint work with Antal Balog. The main result has been obtained independently by Sergei Konyagin.

Daniel Appel, On the abelianization of standard congruence subgroups of the automorphism group of the rank two free group

For an epimorphism pi of the rank two free group F_{2} onto a finite group G write Gamma(G,pi) for the group of all automorphisms f of F_{2} for which pi*f = pi. This is called the standard congruence subgroup of Aut(F) associated to G and pi.

Congruence subgroups associated to abelian groups are closely connected to certain congruence subgroups of SL(2,Z). I will explain this connection and show how to use it to determine the abelianization of Gamma(G,pi) for abelian G.

If time allows, I will also point out some open problems about the abelianization of Gamma(G,pi) for arbitrary finite groups G.

Boris Bukh, Sum-product theorems for polynomials

Suppose A is a set of numbers and f(x,y) is a polynomial, how small can f(A,A) be? If f(x,y)=x+y or f(x,y)=xy, then f(A,A) can be very small indeed if A is aprogression. However, Erdős and Szemerédi proved that A+A and AA cannot be simultaneously small when A is a set of real numbers. In this talk, I will survey this and related results, and will discuss several new results for other polynomial functions f. Joint work with Jacob Tsimerman.

Alexander Stasinski, Deligne-Lusztig theory and generalisations".

We give a gentle introduction to classical Deligne-Lusztig theory of representations of certain linear groups over finite fields by way of examples. We then go on to sketch some recent generalisations of this to linear groups over finite local rings.

Vicky Neale: Bracket quadratics as bases for the integers

One of the classical problems of additive number theory, known as Waring’s problem, is to show that the kth powersform a basis for the integers. That is, for any k there is some s = s(k) such that every positive integer is a sum of s kth powers. Lagrange’s theorem, which says that every positive integer is a sum of four squares, is a special case of this. Waring’s problem was ﬁrst solved by Hilbert, and then a few years later Hardy and Littlewood supplied a new proof, using what is now known as their circle method.

I shall describe how to use a new variation of the circle method to show a Waring-type result: that the bracket quadratics n[n root 2] form an asymptotic basis for the integers. That is, there is some s so that every sufficiently large positive integer is a sum of s numbers of the form n[n root 2]. The proof uses recent work of Green and Tao on the quantitative distribution of polynomial orbits on nilmanifolds. This is joint work with Ben Green.

Amaia Zugadi-Reizabal: Automorphisms of p-adic trees and Hausdorff dimension

In this talk, we will introduce the group of automorphisms of the p-adic tree and show that it is an important source of groups satisfying rare properties. We will focus on the study of the Hausdorff dimension on the group of p-adic automorphisms and we will present some recent results.

**Useful links**:

Campus map (The McCrea Building is number 17)