Forthcoming events in this series


Mon, 20 Feb 2012

16:00 - 17:00
SR1

Kloostermania

Alastair Irving
Mon, 13 Feb 2012

16:00 - 17:00
SR1

An introduction to p-adic cohomology

Jan Tuitman
Abstract

In this talk we will give an introduction to the theory of p-adic (or rigid) cohomology. We will first define the theory for smooth affine varieties, then sketch the definition in general, next compute a simple example, and finally discuss some applications.

Mon, 06 Feb 2012

16:00 - 17:00
SR1

Some Galois groups over Q

Jan Vonk
Abstract

The infamous inverse Galois problem asks whether or not every finite group can be realised as a Galois group over Q. We will see some techniques that have been developed to attack it, and will soon end up in the realms of class field theory, étale fundamental groups and modular representations. We will give some concrete examples and outline the so called 'rigidity method'. 

Mon, 30 Jan 2012

16:00 - 17:00
SR1

The Selberg Class - An Introduction

Daniel Kotzen
Abstract

I will discuss the structure of the Selberg class - in which certain expected properties of Dirichlet series and L-functions are axiomatised - along with the numerous interesting conjectures concerning the Dirichlet series in the Selberg class. Furthermore, results regarding the degree of the elements in the Selberg class shall be explored, culminating in the recent work of Kaczorowski and Perelli in which they prove the absence of elements with degree between one and two.

Mon, 23 Jan 2012

16:00 - 17:00
SR1

On the prime k-tuples conjecture

James Maynard
Abstract

We consider the prime k-tuples conjecture, which predicts that a system of linear forms are simultaneously prime infinitely often, provided that there are no obvious obstructions. We discuss some motivations for this and some progress towards proving weakened forms of the conjecture.

Mon, 21 Nov 2011

16:00 - 17:00
SR1

P-adic L-functions and their special values

Netan Dogra
Abstract

This talk will begin by recalling classical facts about the relationship between values of the Riemann zeta function at negative integers and the arithmetic of cyclotomic extensions of the rational numbers. We will then consider a generalisation of this theory due to Iwasawa, and along the way we shall define the p-adic Riemann zeta function. Time permitting, I will also say something about what zeta values at positive integers have to do with the fundamental group of the projective line minus three points

Mon, 24 Oct 2011

16:00 - 17:00
SR1

Radix conversion for polynomials

Sebastian Pancratz
Abstract

We describe various approaches to the problem of expressing a polynomial $f(x) = \sum_{i=0}^{m} a_i x^i$ in terms of a different radix $r(x)$ as $f(x) = \sum_{j=0}^{n} b_j(x) r(x)^j$ with $0 \leq \deg(b_j) < \deg(r)$. Two approaches, the naive repeated division by $r(x)$ and the divide and conquer strategy, are well known. We also describe an approach based on the use of precomputed Newton inverses, which appears to offer significant practical improvements. A potential application of interest to number theorists is the fibration method for point counting, in current implementations of which the runtime is typically dominated by radix conversions.

Mon, 17 Oct 2011

16:00 - 17:00
SR1

On Maeda's conjecture

Jan Vonk
Abstract

The theory of modular forms owes in many ways lots of its results to the existence of the Hecke operators and their nice properties. However, even acting on modular forms of level 1, lots of basic questions remain unresolved. We will describe and prove some known properties of the Hecke operators, and state Maeda's conjecture. This conjecture, if true, has many deep consequences in the theory. In particular, we will indicate how it implies the nonvanishing of certain L-functions.

Mon, 10 Oct 2011

16:00 - 17:00
SR1

Small Gaps Between Primes

James Maynard
(Oxford)
Abstract

We discuss conjectures and results concerning small gaps between primes. In particular, we consider the work of Goldston, Pintz and Yildrim which shows that infinitely often there are gaps which have size an arbitrarily small proportion of the average gap.

Tue, 21 Jun 2011

14:00 - 15:00
L1

An introduction to integer factorization

Jan Tuitman
(Oxford)
Abstract

(Note change in time and location)

The purpose of this talk is to give an introduction to the theory and

practice of integer factorization. More precisely, I plan to talk about the

p-1 method, the elliptic curve method, the quadratic sieve, and if time

permits the number field sieve.

Mon, 09 May 2011

16:00 - 17:00
SR1

163

Frank Gounelas
(Oxford)
Abstract

I will describe why e^{\pi\sqrt{163}} is almost an integer and how this is related to Q(\sqrt{-163}) having class number one and why n^2-n+41 is prime for n=0,...,39. Bits and pieces about Gauss's class number problem, Heegner numbers, the j-invariant and complex multiplication on elliptic curves will be discussed along the way.

Mon, 31 Jan 2011

16:00 - 17:00
SR1

Rational connectivity and points on varieties

Frank Gounelas
(Oxford)
Abstract

The main aim of this talk will be to present a proof of the Tsen-Lang theorem on the existence of points on complete intersections and sketch a proof of the Grabber-Harris-Starr theorem giving the existence of a section to a fibration of a rationally connected variety over a curve. Time permitting, recent work of de Jong and Starr on rationally simply connected varieties will be discussed with applications to the number theory of hypersurfaces.

Mon, 17 Jan 2011

16:00 - 17:00
SR1

Sums of k-th powers and operators in harmonic analysis

Lillian Pierce
(Oxford)
Abstract

An old conjecture of Hardy and Littlewood posits that on average, the number of representations of a positive integer N as a sum of k, k-th powers is "very small." Recently, it has been observed that this conjecture is closely related to properties of a discrete fractional integral operator in harmonic analysis. This talk will give a basic introduction to the two key problems, describe the  correspondence between them, and show how number theoretic methods, in particular the circle method and mean values of Weyl sums, can be used to say something new in abstract harmonic analysis.

Mon, 29 Nov 2010

16:00 - 17:00
SR1

TBA

Johan Bredberg
(Oxford)