## Plenary Lectures

Michael Atiyah**The impact of physics on mathematics, past, present and future**

In the past few decades there has been a remarkable interaction between physics and mathematics which has had a major impact on areas of pure mathematics that appear far removed from physics. I will review this and put it into historical perspective.

Robert Guralnick**Strongly Dense Subgroups of Algebraic Groups**

Let G be a simple algebraic group. A free nonabelian finitely generated free subgroup H of G is called strongly dense in G if every nonabelian subgroup of H is Zariski dense in G. We will discuss joint work with Breuillard, Green and Tao which shows that strongly dense subgroups exist (over sufficiently large fields) and some recent improvements of this by Brueillard, Guralnick and Larsen.

This has applications to showing that Cayley graphs of finite simple groups of Lie type are expanders and some results on generation of finite simple groups of Lie type. Using these ideas, we can also improve results of Borel and Deligne-Sullivan related to the Hausdorff-Banach-Tarski paradox. The method should also be useful in other situations and is related to the more general ideas in proving results about finite simple groups by using ideas from algebraic group theory.

Ngô Bảo Châu**Arithmetic of certain integrable system** [change to programme]

Nigel Hitchin constructed a beautiful integrable system on the moduli space of Higgs bundle. This integrable system turns out to play several important roles in the (geometric) Langlands program. In particular, it can be regarded as the geometric incarnation of the orbital side of the Arthur-Selberg trace formula. Many difficult problems in harmonic analysis related to the trace formula, for instant the fundamental lemma, can be translated into geometric problems on the Hitchin system.

In my lecture, I will explain a geometric problem related to the topology of singular fibers of an abstract integrable system. The solution of this problem has been crucial for the proof of the fundamental lemma.

Persi Diaconis**The Magic of Martin Gardner**

Martin Gardner brought mathematics to life for millions of people from homemakers to professional mathematicians. I will try to explain what he did and how he did it. From Alice in Wonderland, Psychic exposures, bad poetry, the Game of Life, public key cryptography and a thousand other things, his clarity and curiosity are contagious. But, beware–as someone once wrote:

Endre Szemerédi**On subset sums**

Let \(A ⊂ [1,N]\) be a set of integers. We denote by \(S_A\) the collection of partial sums of \(A\),

$$

S_A= \left\{\sum_{x\in B}x:B\subset A\right\}.

$$

For a positive integer \(l\leq |A|\) we denote by \(l^{*}A\) the collection of partial sums of \(l\) elements of \(A\),

$$

l^{*}A= \left\{\sum_{x\in B}x:B\subset A, |B|=l\right\}.

$$

We will discuss the structure of \(l^{∗}A\) and give a tight bound of the size of \(A\) not containing an \(N\) element arithmetic progression. Some of the results are joint with Van Vu the others are joint work with Simao Herdade.

Cédric Villani**From planets to stars to fluids.**

This will be a lecture on the long time behavior of classical mechanical systems.

Claire Voisin**Points, zero cycles, and rationality questions**

A very classical question in algebraic geometry is whether a given smooth projective variety is rational, that is, birationally isomorphic to projective space, or to give necessary criteria for rationality. Assuming we work over the complex numbers, this is very restrictive on the geometry of the considered variety seen as a complex manifold. Still, if we restrict to the so-called rationally connected varieties, proving that X is non rational is delicate. I will describe classical and more recent methods to approach this problem.

Don Zagier**From finite groups to modular forms**

## Morning Lectures

Tim Austin (Courant Institute, CMI)**Partial difference equations over compact Abelian groups**

Given a compact Abelian group Z, an element z of that group, and a measurable function from it to another Abelian group, one can form a new function by taking the difference of the original function and its translate by z. This is the obvious discrete analog of differentiation, and defines an operator on functions called a differencing operator.

Certain `higher-order' partial difference equations involving such operators arise naturally in pursuing a higher-dimensional analog of Gowers' approach to Szemeredi's Theorem. Given several elements of Z, one asks for a description of those functions on Z which vanish when one applies all of the resulting differencing operators. It turns that as the order of the difference equation increases, one can find surprisingly rich families of solutions. This amounts to a first step towards understanding the inverse problem for the `directional Gowers norms' relevant to the higher-dimensional Szemeredi Theorem.

Christine Bachoc (Bordeaux)**Convex optimization, Fourier analysis and extremal problems in Euclidean geometry**

How densely can unit balls be packed in Euclidean space? How large is the kissing number in dimension n? How many colors are needed to color \({\mathbb R}^n\) so that points at distance one receive different colors?

In this talk, we will review methods giving partial answers to these classical questions through a combination of convex optimization techniques and Fourier analysis on groups. We will explain the links with now well-understood tools in graph theory: namely Lovász theta number and hierarchies of semidefinite programs for the independence number of a graph.

Alexandre Borovik (Manchester)**Black box algebra**

Some natural problems in computational algebra and cryptography require analysis of finite algebraic structures (for example, groups, associative and Lie rings, fields) up to homomorphisms computable by randomised algorithms working in probabilistic polynomial time. This opens a fascinating new chapter of algebra. Some model-theoretic ideas are involved, too, because of the need to construct, by randomised algorithms, some structures (for example, fields) within structures of different nature (for example, groups).

The talk will outline a few basic concepts of this new and beautiful theory.

This is a joint work with Şükrü Yalçınkaya.

Martin Bridson (Oxford)**Capturing infinite groups by their finite quotients**

There are many situations in geometry and group theory where it is natural or convenient to study infinite groups via their finite quotients and finite-index groups. If a group G is residually finite (ie every element survives in some finite quotient) then one might hope to recover a lot of information about the group from the totality of its finite quotients -- equivalently, its pro-finite completion. But precisely which properties of G can be detected in this way, and which cannot? To what extent is a residually-finite group determined by its pro-finite completion or representation theory?

In this lecture I'll survey the recent activity around questions such as these and explain how it connects to other areas of mathematics, particularly geometry.

I'll also explain why there is no algorithm that, given a finitely presented group, can determine if the group has a non-trivial finite quotient.

The lecture will be accessible to a general mathematical audience.

Toby Gee (Imperial College)**The p-adic Langlands program**

The p-adic Langlands program has developed over the last decade to become a central area of number theory in its own right, with links both to the classical Langlands program and to p-adic Hodge theory and the theory of p-adic automorphic forms. It is, however, even conjecturally far from clear what the eventual form of the results should be. I will attempt to explain what is currently known and conjectured, without assuming any knowledge of the Langlands program, p-adic Hodge theory or p-adic automorphic forms.

Harald Helfgott (ENS Paris)**The ternary Goldbach conjecture**

The ternary Goldbach conjecture (1742) asserts that every odd number greater than 5 can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant C satisfies the conjecture. In the years since then, there has been a succession of

results reducing C, but only to levels much too high for a verification by computer up to C to be possible (\(C > 10^{1300}\)). (Works by Ramare and Tao have solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas in the proof.

Daniela Kühn (Birmingham)**Hamilton decompositions of graphs and digraphs**

A Hamilton cycle in a graph or directed graph G is a cycle that contains all the vertices of G. As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of

intensive research, and there many exciting open problems in this area. In this talk I will survey recent results on Hamilton decompositions, i.e. decompositions of the edge-set of a graph or digraph into edge-disjoint Hamilton cycles. Examples include the proofs of a conjecture of Kelly from 1968, which states that every regular tournament has a Hamilton decomposition, and of the so-called Hamilton decomposition conjecture by Nash-Williams from 1970. The techniques involve quasi-random decompositions and expansion (joint work with Bela Csaba, Allan Lo, Deryk Osthus and Andrew Treglown).

Vladimir Markovic (Cambridge)**The Surface Subgroup Problem**

The surface subgroup problem asks whether a given group contains a subgroup that is isomorphic to the fundamental group of a closed surface. In this talk I will survey the role that the surface subgroup problem plays in some important solved and unsolved problems in the theory of 3-manifolds, the geometric group theory, and the theory of arithmetic manifolds.

Corinna Ulcigrai (Bristol)**Polygonal billiards, flows on surfaces and Teichmueller dynamics**

A notable example of a mathematical polygonal billiard is the Ehrenfest model from 1912, in which a point particle bounces off the walls of a periodic planar array of rectangular scatterers. The study of chaotic properties of billiard trajectories in bounded polygons and the related area-preserving flows on compact surfaces has been a topical and successful area of research for the past 30 years. Only very recently though, breakthroughs on our understanding of the Ehrenfest model and other infinite periodic billiards (which correspond to infinite periodic surfaces) were achieved. We will survey some results in this area and we will try to heuristically explain the beautiful connection with the dynamics of the geodesic flow on a space of deformations of geometric stuctures (the Teichmueller flow).

Andrei Yafaev (UCL)**The Andre-Oort conjecture and o-minimality**

Recently Pila and Zannier came up with a strategy for proving the Andre-Oort conjecture using ideas and results from the theory of o-minimality. A lot of progress had already been made on implementing Pila-Zannier ideas. In this talk we will give an overview of the strategy, recent results and their proofs.

## Google Lecture

Johan Granström**Content ID Mathematics**

Videos uploaded to YouTube are scanned against a database of files that have been submitted to us by content owners. This talk is about the mathematics behind the technology that makes this gargantuan matching task possible.