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.

## 12 January 2010

Nina Snaith - Bristol University

##### Random matrix theory and elliptic curves

Abstract

This talk will be an introduction to the connection between random matrix theory and number theory, including an example of the most current research on applications to elliptic curves.

It has been ten years since Katz and Sarnak proposed that, in an appropriate asymptotic limit, zeros of families of L-functions can be modeled by eigenvalues of matrices from the classical compact groups: U(N), O(N) and USp(2N). A few years ago mysterious numerical results of Steven J. Miller showed that in the case of L-functions associated with elliptic curves, anomalous behaviour was observed which did not match the expected O(N) predictions. In this work we propose a solution to this mystery in the form of a modified random matrix ensemble. (Joint with Eduardo Due{\~n}ez, Duc Khiem Huynh, Jon Keating and Steven J. Miller).

## 19 January 2010

Wajid Mannan - University of Southampton

##### Invariant centralized group rings

Abstract

The rational group ring of a finitely generated group can be a large and complicated object (for example it need not be Notherian). I will describe a simplification of the rational group ring, which results in a finite rank Lie algebra over the co-ordinate ring of a finite dimensional rational variety. I will show that the subgroup / normal subgroup structures of a group correspond to the subalgebra / ideal structures of its Lie algebra in the natural way. Finally I will give an example where highly non-trivial information about the normal subgroup structure is preserved in the Lie algebra.

## 26 January 2010

Adrian Tanasa - École Polytechnique

##### Mathematical aspects of quantum field theory on the noncommutative Moyal space

Abstract:

Field theories on the noncommutative Moyal space are introduced; they are manifestly non-local and an appropriate way to implement perturbation theory uses ribbon Feynman graphs. I then overview some recent developments in this field, like the definition of the associated combinatoric Connes-Kreimer Hopf algebra underlying non-local renormalization or the relation with the Bollobas-Riordan topological polynomial of ribbon graphs.

## 02 February 2010

Andrew Booker - Bristol University

##### Alan Turing and the Riemann hypothesis

Abstract:

Many mathematicians are familiar with Alan Turing as a logician, pioneer of computer science, and even war hero. Not so many know that he was also a number theorist. I will describe Turing's interest in the Riemann hypothesis, in a manner accessible to all.

## 09 February 2010 - COLLOQUIUM TALK

Frank Smith - UCL

- please see the Departmental Colloquia webpage

## 16 February 2010

READING WEEK - NO SEMINAR

## 23 February 2010

Malwina Luczak - London School of Economics

##### Glauber dynamics for the Ising model on the complete graph

Abstract:

We study the Glauber dynamics for the Ising model on the complete graph on $n$ vertices, also known as the Curie-Weiss Model, with inverse temperature $\beta/n$. 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 $O(n \log n)$. (Joint work with David Levin and Yuval Peres.)

## 02 March 2010

David Conlon - University of Cambridge

##### Combinatorial theorems in random sets

Abstract:

The famous theorem of Szemerédi says that for any natural number k and any a > 0 there exists n such that if N >= n then any subset A of the set [N] = {1, 2,... , N} of size |A| >= a N contains an arithmetic progression of length k. We consider the question of when such a theorem holds in a random set. More precisely, we say that a set X is (a, k)-Szemerédi if every subset Y of X that contains at least a|X| elements contains an arithmetic progression of length k. Let [N]_p be the random set formed by taking each element of [N] independently with probability p. We prove that there is a threshold at about p = N^{-1/(k-1)} where the probability that [N]_p is (a, k)-Szemerédi changes from being almost surely 0 to almost surely 1.

There are many other similar problems within combinatorics. For example, Turán's theorem and Ramsey's theorem may be relativised, but until now the precise probability thresholds were not known. Our method seems to apply to all such questions, in each case giving the correct threshold. This is joint work with Tim Gowers.

## 09 March 2010 - COLLOQUIUM TALK

John Toland - University of Bath

- please see the Departmental Colloquia webpage

## 16 March 2010

Joe Chuang - City University, London

##### Tiling and tilting

Abstract:

I will describe some elementary combinatorics of tilings of the plane by rhombi and indicate some connections with the representation theory of the symmetric groups. Certain `tiltings' of the tilings are meaningful in representation theory.

## 23 March 2010

Adam Ostaszewski - London School of Economics

##### Kestelman shift compactness

For the abstract of Dr Ostaszewski's talk, please click here

## EXTRA SEMINAR

## FRIDAY 14 MAY 2010 at 12pm - PLEASE NOTE: THIS SEMINAR IS ON A FRIDAY

Günter Ziegler - TU Berlin

##### On the power and failure of topological methods in combinatorics

Abstract:

Topological methods (such as the Borsuk-Ulam Theorem and its extensions) have celebrated a number of remarkable successes -- starting perhaps with Birch's 1959 paper ``On 3N points in a plane'' and then with Lovasz' 1979 resolution of the Kneser conjecture.

The approach has been formalized by efforts of Sarkaria, Zivaljevic, and others, leading to the ``Configuration Space/Text Map'' (CS/TM) scheme.

However, a slightly closer look also reveals very interesting limitations and failures of this method. We will discuss them in terms of the Tverberg Theorem (1966), where the topological method works only in the prime-power case - and the recent Tight Colored Tverberg Theorem (2009), where our topological approach only works in the prime case.

(Joint work with Pavle V. Blagojevic and Benjamin Matschke.)