Mon, 10 Feb 2014

16:00 - 17:00
C5

Diophantine Properties of Nilpotent Lie Groups

Henry Bradford
(Oxford University)
Abstract

A finite set of elements in a connected real Lie group is "Diophantine" if non-identity short words in the set all lie far away from the identity. It has long been understood that in abelian groups, such sets are abundant. In this talk I will discuss recent work of Aka; Breuillard; Rosenzweig and de Saxce concerning this phenomenon (and its limitations) in the more general setting of nilpotent groups. 

Mon, 27 Jan 2014

16:00 - 17:00
C5

Limit-periodic functions and their exponential sums

Eugen Keil
(Oxford University)
Abstract
In the first part of the talk we are going to build up some intuition about limit-periodic functions and I will explain why they are the 'simplest' class of arithmetic functions appearing in analytic number theory. In the second part, I will give an equivalent description of 'limit-periodicity' by using exponential sums and explain how this property allows us to solve 'twin-prime'-like problems by the circle method.
Thu, 13 Feb 2014

16:00 - 17:00
L5

Covering systems of congruences

Bob Hough
(Oxford University)
Abstract
A distinct covering system of congruences is a collection

\[

(a_i \bmod m_i), \qquad 1\ \textless\ m_1\ \textless\ m_2\ \textless\ \ldots\ \textless\ m_k

\]

whose union is the integers. Erd\"os asked whether there are covering systems for which $m_1$ is arbitrarily large. I will describe my negative answer to this problem, which involves the Lov\'{a}sz Local Lemma and the theory of smooth numbers.

Mon, 20 Jan 2014

16:00 - 17:00
C5

The private life of Bryan

Jan Vonk
(Oxford University)
Abstract

This talk will discuss the discovery of Heegner points from a historic perspective. They are a beautiful application of analytic techniques to the study of rational points on elliptic curves, which is now a ubiquitous theme in number theory. We will start with a historical account of elliptic curves in the 60's and 70's, and a correspondence between Birch and Gross, culminating in the Gross-Zagier formula in the 80's. Time permitting, we will discuss certain applications and ramifications of these ideas in modern number theory. 

Mon, 03 Feb 2014

16:00 - 17:00
C5

"Moat lemmas" and mean values of exponential sums

Simon Myerson
(Oxford University)
Abstract

In 1997 V. Bentkus and F. Götze introduced a technique for estimating $L^p$ norms of certain exponential sums without needing an explicit estimate for the exponential sum itself. One uses instead a kind of estimate I call a "moat lemma". I explain this term, and discuss the implications for several kinds of point-counting problem which we all know and love.

Wed, 20 Nov 2013
10:30
Queen's College

Introduction to limit groups

Montserrat Casals
(Oxford University)
Abstract
In this talk I will introduce the class of limit groups and discuss its characterisations from several different perspectives: model-theoretic, algebraic and topological. I hope that everyone will be convinced by at least one of the approaches that this class of groups is worth studying.
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