All seminars take place on Tuesdays at 4.00 pm in Room 500 on the 5th floor of the Mathematics Dept. See how to find us for further details. There will be tea afterwards in room 606.

If you require any more information on the Pure seminars please contact Dr Nadia Sidorova e-mail: n.sidorova AT ucl.ac.uk or tel: 020-7679-7864. If you would like to receive weekly announcements including titles and abstracts, you are welcome to join the seminar mailing list.

## 06 October 2009

Jens Funke - University of Durham

##### Quadratic Forms and Modular Forms

Abstract

In this talk, we explore the relationship between integral quadratic forms, both positive definite and indefinite, to modular forms. Modular forms play an increasingly central role in modern number theory. We give an introduction to the subject concentrating on the case of quadratic forms with 3 variables. Among the topics we discuss are class numbers of imaginary quadratic fields, representation numbers for the sum of three squares, and the values of the famous j-invariant at quadratic irrationalities in the upper half plane.

## 13 October 2009

Florian Pop - University of Pennsylvania

##### Isomorphism versus elementary equivalence

Abstract

I plan to speak about a well known (still) open problem in arithmetic geometry and model theory, which concerns defining the isomorphism of geometrically and/or arithmetically significant function fields in a first order way. I will explain what is known and the difficulties in trying to completely solve the problem.

## 20 October 2009

David Loeffler - University of Cambridge

##### Modular forms from a p-adic viewpoint

Abstract:

Modular forms are objects of tremendous importance in many branches of modern number theory, which is surprising as their definition appears to belong solidly to the world of complex analysis. One aspect of this is a form of "p-adic continuity" for a prime p: given two weights that are highly congruent modulo p, in many cases there exist congruences modulo high powers of p between modular forms at those weights.

In the talk I'll begin by explaining the definition of modular forms, and go on to describe some of the ideas that go into studying their p-adic behaviour. If time permits I will also introduce automorphic forms, which are a generalisation of modular forms, and mention current research from which a p-adic theory of automorphic forms is beginning to emerge.

## 27 October 2009

Roger Heath-Brown - University of Oxford

##### The distribution of fractional parts of sequences $\alpha n^2$

Abstract:

For typical irrational values of $\alpha$ the sequence of fractional parts of $\alpha n^2$ over short intervals appears to have a Poisson distribution. We will discuss some open questions and recent work on this topic.

## 03 November 2009

Tadashi Tokieda - University of Cambridge

##### Mathematics on paper

Abstract:

A lot of mathematics can be done on paper---literally on paper, i.e. almost 2-dimensional (`almost' because it also partakes of 3D properties), almost elastic (`almost' because it also has memory) medium. From topology via algebra and geometry to nonlinear PDEs, we will meet many phenomena involving paper with table-top demos, some of them well-known, but most

of them hopefully unfamiliar to the audience. The talk is accessible to undergraduates, though it broaches hard open problems. And we promise at least one new, nice theorem you can take home.

## 10 November 2009

READING WEEK - NO SEMINAR

## 17 November 2009

Iskander Aliev - Cardiff University

##### Feasibility of Integer Knapsacks: Average Behavior of the Frobenius

Numbers

Abstract

The largest integer which cannot be represented as a non-negative integral combination of given set of positive integers is called the Frobenius number of these integers. We show that for large instances the order of magnitude of the expected Frobenius number is given by its lower bound.

## 24 November 2009

Istvan Gyongy - University of Edinburgh

##### Cauchy problems with periodic controls

Abstract:

Cauchy problems for linear evolution equations are considered, which include second order parabolic PDEs and symmetric hyperbolic system of PDEs. A class of numerical methods for these problems is interpreted as periodic controls injected in the equations. An expansion in powers of p, the period of the controls, is obtained for the solution of the controlled equations. Applications to numerical solutions of PDEs and nonlinear ODEs are presented. In particular, it is shown that the order of accuracy of finite difference schemes and splitting-up approximations can be made as high as wanted by an implementation of Richardson's idea. The talk is based on joint results

with Nicolai Krylov.

## 01 December 2009

Kirill Cherednichenko - Cardiff University

##### Two-scale $\Gamma$-convergence and homogenisation of degenerate nonlinear PDE

Abstract:

The talk will begin with an outline of the 'mathematical theory of homogenisation', which studies PDE with rapidly oscillating data. The classical results in this area (from the early 1970s) concern second-order PDE that are uniformy elliptic, and the related theory is well developed, including nonlinear settings via the so-called $\Gamma$-convergence for variational integrals (De Giorgi, Dal Maso, Braides). It can be shown however, that introducing a high degree of contrast into such problems, thus violating the assumption of uniform ellipticity, may result in homogenised limits of a non-standard (or ``non-classical'') kind.

The main focus of the talk will be on nonlinear (and in general, non-convex) variational problems with high contrast. In order to pass to the homogenisation limit in such problems, we develop the concept of ``two-scale $\Gamma$-convergence'', which may be thought of as a hybrid of the classical $\Gamma$-convergence mentioned above and the two-scale convergence (Allaire,

Briane, Zhikov). I will demonstrate the need for such a tool by showing that in the high-contrast case the minimising sequences may be non-compact in the $L^p$ space and the corresponding minima may not converge to the minimum of the usual $\Gamma$-limit. I will discuss a compactness principle for high-contrast functionals with respect to the two-scale $\Gamma$-convergence, which in particular implies convergence of their minima. This is joint work with Mikhail Cherdantsev.

## 08 December 2009

NO SEMINAR (due to Brian Davies 65th birthday conference at King's College)

## 15 December 2009

Thomas Mueller - Queen Mary, University of London

##### A class of groups universal for free R-tree actions

Abstract

Please click here to open Thomas Mueller's Abstract document