All seminars take place on Tuesdays at 4.00 pm in Room 505 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.

## 26 October 2010 - COLLOQUIUM TALK

John Ball - Oxford University

- please see the Departmental Colloquia webpage

## 02 November 2010

Javier López Peña - UCL

##### The field with one element: an object in search of a definition

Abstract:

In the recent years there has been a burst of approaches dealing with the problem of developing a notion of algebraic geometry and theory of motives over a mysterious "field of characteristic one". In this talk we will present some of the original motivations that lead to this search and outline the principal ideas on the main different approaches to the problem, with a particular mention to our own geometrical approach via torified varieties, schemes admitting a stratification by split tori that permits to give an F_1 structure and easily compute the corresponding zeta function.

## 09 November 2010

READING WEEK - NO SEMINAR

## 16 November 2010

Jonathan Bennet - University of Birmingham

##### Classical and nonlinear Brascamp-Lieb inequalities and applications in harmonic analysis

Abstract:

The Brascamp-Lieb inequalities simultaneously generalise important classical inequalities in euclidean analysis such as the multilinear H\"older, Young's convolution and Loomis-Whitney inequalities. In the first part of this talk we will describe a recent and elementary

approach to these inequalities based on heat flow. In the second part we will investigate some "nonlinear" generalisations which arise quite snaturally in harmonic analysis and partial differential equations.

## 23 November 2010

Jacqueline Fleckinger - Université Toulouse 1

##### On fundamental positivity and negativity for some indefinite problems defined on the whole space

Abstract:

We consider the problem $-\Delta u = \lambda g u + f$ on $\mathbb{R}^N$, $N\ge 3$. For functions $g$ and $f$ satisfying some growth conditions, we compare the solution $u$ with the groundstate.

## 30 November 2010

Lillian Pierce - Oxford University

##### Number Theoretic Methods for Discrete Analogues in Harmonic Analysis

Abstract:

Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space (easy). Counting the number of representations of an integer N as a sum of k squares (historically hard). In this talk we'll survey a menagerie of discrete analogues of operators arising in harmonic analysis, including singular integral operators (such as the Hilbert transform), maximal functions, and fractional integral operators. In certain cases we can learn everything we want to know about the discrete operator immediately, from its continuous analogue. In other cases the discrete operator requires a completely new approach. We'll see what makes a discrete operator easy or hard to treat, and outline some of the methods that are breaking new ground, key aspects of which come from number theory. In particular, we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood. No specialist knowledge of singular integral operators or the circle method will be assumed.

## 07 December 2010

Ostap Hryniv - Durham University

##### Phase transition in a model of microtubule growth

Abstract:

We study a continuous time stochastic process on strings made of two types of particles, whose dynamics mimics the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterisation of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

## 14 December 2010 - COLLOQUIUM TALK

Brian Davies - King's College London