Forthcoming events in this series


Thu, 18 Jun 2009

16:00 - 17:00
SR1

An Introduction to Tauberian Theorems

Timothy Trudgian
(Mathematical Institute, Oxford)
Abstract

Suppose a power series $f(x):= \sum_{n=0}^{\infty} a_{n} x^{n}$ has radius of convergence equal to $1$ and that $lim_{x\rightarrow 1}f(x) = s$. Does it therefore follow that $\sum_{n=0}^{\infty} a_{n} = s$? Tauber's Theorem answers in the affirmative, \textit{if} one imposes a certain growth condition (a \textit{Tauberian Condition}) on the coefficients $a_{n}$. Without such a condition it is clear that this cannot be true in general - take, for example, $f(x) = \sum_{n=0}^{\infty} (-1)^{n} x^{n}.$

Mon, 01 Jun 2009

16:00 - 17:00
SR1

Introduction to the Birch--Swinnerton-Dyer Conjecture. III: Average ranks, the Artin--Tate conjecture and function fields.

George Walker
(Mathematical Insitute, Oxford)
Abstract

In the previous talks we have seen the formulation of the Birch--Swinnerton-Dyer conjecture. This talk will focus on a fundamental question in diophantine geometry. Namely, given an algebraic curve \textit{C} defined over $\mathbb{Q}$ possessing at least one rational point, what is

the probability that \textit{C} has infinitely many rational points?

For curves of genus 0, the answer has been known ever since the ancient Greeks roamed the earth, and for genus > 1 the answer is also known (albeit for a much shorter time). The remaining case is genus 1, and this question has a history filled with tension and

conflict between data and conjecture.

I shall describe the heuristics behind the conjectures, taking into account the Birch--Swinnerton-Dyer Conjecture and the Parity Conjecture. I shall go on to outline the contrary numeric data, both in families of elliptic curves and for all elliptic curves of increasing conductor.

If one instead considers elliptic curves over function fields $\mathbb{F}_{q} (t)$, then, via a conjecture of Artin and Tate, one can compute the rank (and more) of elliptic curves of extremely large discriminant degree. I shall briefly describe the interplay between Artin--Tate and

Birch--Swinnerton-Dyer, and give new evidence finally supporting the conjecture.

Mon, 09 Mar 2009

16:00 - 17:00
SR1

The Chevalley-Warning Theorem

Dr Damiano Testa
(The Mathematical Institute, Oxford)
Abstract

The goal of this talk is to give sufficient conditions for the existence of points on certain varieties defned over finite fields.

Mon, 02 Mar 2009

16:00 - 17:00
SR1

Classical Primality Testing

Sebastian Pancratz
(Mathematical Institute, Oxford)
Abstract

This talk will mention methods of testing whether a given integer is prime. Included topics are Carmichael numbers, Fermat and Euler pseudo-primes and results contingent on the Generalised Riemann Hypothesis.

Mon, 23 Feb 2009

16:00 - 17:00
SR1

Ostrowski's Theorem and other diversions

Jahan Zahid
(Oxford)
Abstract

Aside from a few tangential problems, this seminar will include a proof of Ostrowski's Theorem. This states than any norm over the rationals is equivalent to either the Euclidean norm or the $p$-adic norm, for some prime $p$.

Mon, 09 Feb 2009

16:00 - 17:00
SR1

Dirichlet's Approximation Theorem

Johan Bredberg
(Oxford)
Abstract

This talk will introduce Dirichlet's Theorem on the approximation of real numbers via rational numbers. Once this has been established, a stronger version of the result will be proved, viz Hurwitz's Theorem.

Mon, 02 Feb 2009

16:00 - 17:00
SR1

Jensen's Theorem and a Simple Application

Timothy Trudgian
(Mathematical Institute Oxford)
Abstract

This second 'problem sheet' of the term includes a proof of Jensen's Theorem for the number of zeroes of an analytic function in a disc, the usefulness of which is highlighted by an application to the Riemann zeta-function.

Mon, 01 Dec 2008

16:00 - 17:00
SR1

A Combinatorial Approach to Szemer\'{e}di's Theorem on Arithmetic Progressions

Sebastian Pancratz
(University of Oxford)
Abstract
This talk will give detailed proofs of Szemer\'{e}di's Regularity Lemma for graphs and the deduction of Roth's Theorem. One can derive Szemer\'{e}di's Theorem on arithmetic progressions of length $k$ from a suitable regularity result on $(k-1)$-uniform hypergraphs, and this will be introduced, although not in detail.
Mon, 28 Jan 2008

15:00 - 16:00
SR1

Some mathematics in musical harmonics

Tim Trudgian
(Mathematical Insitute, Oxford)
Abstract

A brief overview of consonance by way of continued fractions and modular arithmetic.

Mon, 19 Nov 2007

15:00 - 16:00
SR1

A digression from the zeroes of the Riemann zeta function to the behaviour of $S(t)$

Tim Trudgian
(Mathematical Insitute, Oxford)
Abstract

Defined in terms of $\zeta(\frac{1}{2} +it)$ are the Riemann-Siegel functions, $\theta(t)$ and $Z(t)$. A zero of $\zeta(s)$ on the critical line corresponds to a sign change in $Z(t)$, since $Z$ is a real function. Points where $\theta(t) = n\pi$ are called Gram points, and the so called Gram's Law states between each Gram point there is a zero of $Z(t)$, and hence of $\zeta(\frac{1}{2} +it)$. This is known to be false in general and work will be presented to attempt to quantify how frequently this fails.