Forthcoming events in this series


Mon, 16 Jun 2014

16:00 - 17:00
C5

A Hitchhiker's guide to Shimura Varieties

Tom Lovering
(Harvard University)
Abstract

Since their introduction, Shimura varieties have proven to be important landmarks sitting right at the crossroads between algebraic geometry, number theory and representation theory. In this talk, starting from the yoga of motives and Hodge theory, we will try to motivate Deligne's construction of Shimura varieties, and briefly survey some of their zoology and basic properties. I may also say something about the links to automorphic forms, or their integral canonical models.

Mon, 09 Jun 2014

16:00 - 17:00
C5

Intersections of progressions and spheres

Sean Eberhard
(University of Oxford)
Abstract

We state a conjecture about the size of the intersection between a bounded-rank progression and a sphere, and we prove the first interesting case, a result of Chang. Hopefully the full conjecture will be obvious to somebody present.

Mon, 02 Jun 2014

16:00 - 17:00
C5

Isogeny pancakes

Chloe Martindale
(Leiden University)
Abstract

Pancakes.

Mon, 26 May 2014

16:00 - 17:00
C5

An attempt to find the optimal constant in Balog-Szemeredi-Gowers theorem.

Przemysław Mazur
(University of Oxford)
Abstract

The Balog-Szemeredi-Gowers theorem states that, given any finite subset of an abelian group with large additive energy, we can find its large subset with small doubling constant. We can ask how this constant depends on the initial additive energy. In the talk, I will give an upper bound, mention the best existing lower bound and, if time permits, present an approach that gives a hope to improve the lower bound and make it asymptotically equal to the upper bound from the beginning of the talk.

Mon, 19 May 2014

16:00 - 17:00
C5

Periods of Hodge structures and special values of the gamma function

Javier Fresán
(Max Planck Institute Bonn)
Abstract

At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are always products of values of the gamma function at rational numbers, with exponents determined by the Hodge decomposition. I will explain a proof of an alternating variant of this conjecture for the cohomology groups of smooth, projective varieties over the algebraic numbers acted upon by a finite order automorphism.

Mon, 12 May 2014

16:00 - 17:00
C5

TBA

Frederick Manners
(University of Oxford)
Mon, 05 May 2014

16:00 - 17:00
C5

How common are solutions to equations?

Simon Myerson
(University of Oxford)
Abstract

Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong.

I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.

Mon, 10 Mar 2014

16:00 - 17:00
C5

TBA

Miguel Walsh
(Oxford University)
Mon, 03 Mar 2014

16:00 - 17:00
C5

The elliptic curve discrete logarithm problem

Christophe Petit
Abstract

The elliptic curve discrete logarithm problem (ECDLP) is commonly believed to be much harder than its finite field counterpart, resulting in smaller cryptography key sizes. In this talk, we review recent results suggesting that ECDLP is not as hard as previously expected in the case of composite fields.

We first recall how Semaev's summation polynomials can be used to build index calculus algorithms for elliptic curves over composite fields. These ideas due to Pierrick Gaudry and Claus Diem reduce ECDLP over composite fields to the resolution of polynomial systems of equations over the base field.

We then argue that the particular structure of these systems makes them much easier to solve than generic systems of equations. In fact, the systems involved here can be seen as natural extensions of the well-known HFE systems, and many theoretical arguments and experimental results from HFE literature can be generalized to these systems as well.

Finally, we consider the application of this heuristic analysis to a particular ECDLP index calculus algorithm due to Claus Diem. As a main consequence, we provide evidence that ECDLP can be solved in heuristic subexponential time over composite fields. We conclude the talk with concrete complexity estimates for binary curves and perspectives for furture works.

The talk is based on joint works with Jean-Charles Faugère, Timothy Hodges, Yung-Ju Huang, Ludovic Perret, Jean-Jacques Quisquater, Guénaël Renault, Jacob Schlatter, Naoyuki Shinohara, Tsuyoshi Takagi

Mon, 24 Feb 2014

16:00 - 17:00
C5

Solving equations

Bryan Birch
(Oxford University)
Mon, 17 Feb 2014

16:00 - 17:00
C5

The trace formula

Benjamin Green
(Oxford University)
Abstract

In this talk I will explain the basic motivation behind the trace formula and give some simple examples. I will then discuss how it can be used to prove things about automorphic representations on general reductive groups.

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, 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.

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.

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, 02 Dec 2013

17:00 - 18:00
C5

The pyjama problem

Freddie Manners
Abstract

The 'pyjama stripe' is the subset of the plane consisting of a vertical

strip of width epsilon about every integer x-coordinate. The 'pyjama

problem' asks whether finitely many rotations of the pyjama stripe about

the origin can cover the plane.

I'll attempt to outline a solution to this problem. Although not a lot

of this is particularly representative of techniques frequently used in

additive combinatorics, I'll try to flag up whenever this happens -- in

particular ideas about 'limit objects'.

Mon, 25 Nov 2013

17:00 - 18:00
C5

Obstructions to the Hasse principle

Francesca Balestrieri
Abstract

This talk will be a gentle introduction to the main ideas behind some of the obstructions to the Hasse principle. In particular, I'll focus on the Brauer-Manin obstruction and on the descent obstruction, and explain briefly how other types of obstructions could be constructed.

Mon, 18 Nov 2013

17:00 - 18:00
C5

Artin's conjecture on p-adic forms

Jan Dumke
Abstract

In the 1930's E. Artin conjectured that a form over a p-adic field of degree d has a non-trivial zero whenever n>d^2. In this talk we will discuss this relatively old conjecture, focusing on recent developments concerning quartic and quintic forms.

Mon, 11 Nov 2013

17:00 - 18:00
C5

Cubic polynomials represented by norm forms

Alastair Irving
Abstract

I will describe how a sieve method can be used to establish the Hasse principle for the variety

$$f(t)=N(x_1,\ldots,x_k),$$

where $f$ is an irreducible cubic and $N$ is a norm form for a number field satisfying certain hypotheses.

Mon, 28 Oct 2013

17:00 - 18:00
C5

Mixed Motives in Number Theory

Netan Dogra
Abstract

Mixed motives turn up in number theory in various guises. Rather than discuss the rather deep foundational questions involved, this talk will aim

to give several illustrations of the ubiquity of mixed motives and their realizations. Along the way I hope to mention some of: the Mordell-Weil

theorem, the theory of height pairings, special values of L-functions, the Mahler measure of a polynomial, Galois deformations and the motivic

fundamental group.

Mon, 21 Oct 2013

17:00 - 18:00
C5

Finding Galois Representations

Ben Green
Abstract

It is well known that one can attach Galois representations to certain modular forms, it is natural to ask how one might generalise this to produce more Galois representations. One such approach, due to Gross, defines objects called algebraic modular forms on certain types of reductive groups and then conjectures the existence of Galois representations attached to them. In this talk I will outline how for a particular choice of reductive group the conjectured Galois representations exist and are the classical modular Galois representations, thus providing some evidence that this is a good generalisation to consider.

Mon, 14 Oct 2013

17:00 - 18:00
C5

Calculations with elliptic curves

Jan Vonk
Abstract

We will discuss some geometric methods to study Diophantine equations. We focus on the case of elliptic curves and their natural generalisations: Abelian varieties, Calabi-Yau manifolds and hyperelliptic curves.