Forthcoming events in this series


Mon, 22 Apr 2013

16:00 - 17:00
SR1

The eigencurve

Jan Vonk
(Oxford)
Mon, 04 Mar 2013

16:00 - 17:00
SR1

A primer on Burgess bounds

Lillian Pierce
(Oxford)
Abstract

We'll present a proof of the basic Burgess bound for short character sums, following the simplified presentation of Gallagher and Montgomery.

Mon, 11 Feb 2013

16:00 - 17:00
SR1

TBC

Netan Dogra
(Oxford)
Mon, 26 Nov 2012

16:00 - 17:00
SR1

Once Upon a Time in Egypt: How the Story of Rational Points Began

Simon Myerson
(Oxford)
Abstract

A nice bed-time story to end the term. It is often said that ideas like the group law or isogenies on elliptic curves were 'known to Fermat' or are 'found
in Diophantus', but this is rarely properly explained. I will discuss the first work on rational points on curves from the point of view of modern number
theory, asking if it really did anticipate the methods we use today.

Mon, 15 Oct 2012

16:00 - 17:00
SR1

Simultaneous prime values of pairs of quadratic forms

Lillian Pierce
(Oxford)
Abstract

Given a form $F(x)$, the circle method is frequently used to provide an asymptotic for the number of representations of a fixed integer $N$ by $F(x)$. However, it can also be used to prove results of a different flavor, such as showing that almost every number (in a certain sense) has at least one representation by $F(x)$. In joint work with Roger Heath-Brown, we have recently considered a 2-dimensional version of such a problem. Given two quadratic forms $Q_1$ and $Q_2$, we ask whether almost every integer (in a certain sense) is simultaneously represented by $Q_1$ and $Q_2$. Under a modest geometric assumption, we are able to prove such a result if the forms are in $5$ variables or more. In particular, we show that any two such quadratic forms must simultaneously attain prime values infinitely often. In this seminar, we will review the circle method, introduce the idea of a Kloosterman refinement, and investigate how such "almost all" results may be proved.


Mon, 28 May 2012

16:00 - 17:00
SR1

The congruent number problem

Frank Gounelas
Abstract

Which positive integers are the area of a right angled triangle with rational sides? In this talk I will discuss this classical problem, its reformulation in terms of rational points on elliptic curves and Tunnell's theorem which gives a complete solution to this problem assuming the Birch and Swinnerton-Dyer conjecture.

Mon, 07 May 2012

16:00 - 17:00
SR1

p-adic zeta functions, p-adic polylogarithms and fundamental groups

Netan Dogra
Abstract

This talk will attempt to say something about the p-adic zeta function, a p-adic analytic object which encodes information about Galois cohomology of Tate twists in its special values. We first explain the construction of the p-adic zeta function, via p-adic Fourier theory. Then, after saying something about Coleman integration, we will explain the interpretation of special values of the p-adic zeta function as limiting values of p-adic polylogarithms, in analogy with the Archimedean case. Finally, we will explore the consequences for the de Rham and etale fundamental groupoids of the projective line minus three points.

Mon, 30 Apr 2012

16:00 - 17:00
SR1

Vinogradov's Three Prime Theorem

James Maynard
Abstract

Vinogradov's three prime theorem resolves the weak Goldbach conjecture for sufficiently large integers. We discuss some of the ideas behind the proof, and discuss some of the obstacles to completing a proof of the odd goldbach conjecture.