# Mathematics Colloquium 2011-12 Abstracts

7 October 2011 **Martin Hairer **(Warwick) *Solving the KPZ equation*

Abstract: The KPZ equation was originally introduced in the eighties as a model of surface growth, but it was soon realised that its solution is a "universal" object describing the crossover between the Gaussian universality class and the KPZ universality class. The mathematical proof of its universality however is still an open problem, in particular because of the lack of a good approximation theory for the equation. Indeed, the only known way so far to mathematically interpret solutions to the KPZ equation is to reduce it to a linear stochastic PDE via a non-linear transformation called the Cole-Hopf transform. Unfortunately, the resulting linear equation does itself lack a good approximation theory and many microscopic models do not behave well under the Cole-Hopf transform.

In this talk, we present a new notion of solution to the KPZ equation that bypasses the use of the Cole-Hopf transform. Our approach also allows to factorise the solution map into a "universal" (i.e. independent of initial condition) measurable map, composed with a solution map with good continuity properties. This lays the foundations for a robust approximation theory to the KPZ equation, which is needed to prove its universality. As a byproduct of the construction, we obtain very detailed regularity estimates on the solutions, as well as a new homogenisation result.

14 October 2011 **Inna Capdeboscq **(Warwick) *Finite Simple Groups of medium size*

Abstract: In this talk we will discuss some recent work related to the Generation-2 proof of Classification of Finite Simple Groups

21 October 2011 **Jonathan Bennett **(Birmingham) *The restriction problem for the Fourier transform** *

Abstract: The classical and far-reaching restriction conjecture for the Fourier transform concerns the size of the restriction of the Fourier transform of a function to a curved submanifold of Euclidean space. In the first part of this talk we give an introduction to this conjecture, including its celebrated connection with the Kakeya problem from combinatorial geometry. In the second part we describe a revealing multilinear perspective which has led to a very recent breakthrough in our understanding of Fourier restriction phenomena.

28 October 2011 **Trevor Wooley **(Bristol)* Vinogradov's mean value theorem, function fields, and moduli spaces*

Abstract: Exponential sums of large degree play a prominent role in the analysis of problems spanning the analytic theory of numbers. In 1935, I.

M. Vinogradov devised a method for estimating their mean values very much more efficient than the methods available hitherto due to Weyl and van der Corput, and subsequently applied his new estimates to investigate the zero-free region of the Riemann zeta function, in Diophantine approximation, and in Waring’s problem. Over the past 75 years, estimates for the moments underlying Vinogradov’s mean value theorem have failed to achieve those conjectured by a factor of roughly log k in the number of implicit variables required to successfully analyse exponential sums of degree k.

Our goal in this talk is to introduce Vinogradov's mean value theorem, and to explain the author's recent work that comes within a stone’s throw of the best possible conclusions. It transpires that the new methods are sufficiently robust so as to be applicable in function fields (work joint with Yu-Ru Liu). By means of an idea presented by Ellenberg and Venkatesh, our work has implications for the geometry of moduli spaces.

4 November 2011 **Oleg Pikhurko **(Warwick) *Graph Limits*

Abstract: We discuss main ideas and results about limits of dense graphs, concentrating on the Rademacher-Turan problem that asks for the smallest number of triangles that a graph of given order and size can have.

11 November 2011 **Danny Calegari **(Cambridge) *Stable commutator length in free groups*

Abstract: Stable commutator length (scl) answers the question: “what is the simplest surface in a given space with prescribed boundary?” where “simplest” is interpreted in topological terms. This topological deﬁnition is complemented by several equivalent deﬁnitions - in group theory, as a measure of non- commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, scl is concerned with questions such as computing the genus of a knot, or ﬁnding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, scl is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures).

18 November 2011 **Sebastian van Strien **(Warwick) *One-dimensional dynamics: a survey of recent results*

Abstract: Over the last decades there has been a spectacular development of the theory on the dynamics of maps of an interval or a circle. In this talk I will discuss recent results and open problems.

25 November 2011 **Ali Alavi **(Cambridge) *Quantum Chemical Games of Life*

Abstract: We have developed an algorithm to solve the "Full CI problem" of quantum chemistry (aka the electronic Schrodinger equation) using a quantum Monte Carlo approach. The algorithm is of a population dynamics of walkers which inhabit Slater determinant space, and is designed so that the evolving population develops a distribution that matches the exact ground-state eigenvector of the underlying (many-electron) Hamiltonian [1]. We show that this population dynamics has a remarkable emergence characteristic, akin to symmetry-breaking phase transitions in classical statistical mechanical systems.

The sign problem posed by Hamiltonian matrix elements of positive and negative sign is solved through a combination of walker annihilation and a "survival of the fittest" criterion [2] (the latter greatly reducing the dependence of the algorithm on walker annihilation). We show that the method can be used to solve problems with FCI accuracy in astronomically large Hilbert spaces.

We will give examples of the algorithm at work in real systems in sizeable basis sets, ranging from atoms, anions and diatomic molecules. Finally, we discuss the scaling of the algorithm with the number of electrons and basis functions [3].

[1] G.H. Booth, A.J.W. Thom and Ali Alavi, J. Chem. Phys., 131 , 054106, (2009).

[2] Deidre Cleland, G.H. Booth, and Ali Alavi, J. Chem. Phys., 132 , 041103, (2010).

[3] Deidre Cleland, George Booth, and Ali Alavi, J. Chem. Phys., 134 , 024112, (2011)

2 December 2011 **Darren Crowdy **(Imperial) *Solving problems in multiply connected domains*

Abstract: Motivated by problems arising in the applied sciences, this talk surveys a new theoretical approach to solving problems in multiply connected planar domains as developed by the speaker (and his group) in recent years. Multiply connected domains are ``regions with holes'' and are ubiquitous in applications; whenever two or more objects/entities interact in some ambient medium the analysis may call for the methods discussed in this lecture. We will advocate the use of ideas from constructive function theory and complex analysis to provide quasi-analytical solutions to such problems in terms of the so-called ``Schottky-Klein prime function'' -- a very important classical special function that is hardly known to non-specialists, but which is relevant to a surprisingly wide range of applied mathematical problems often facilitating concise and elegant representations of their solutions. While the theory of the prime function was first discussed well over a hundred years ago, it has hardly been used in applied problems and it is only recently that numerical methods have been developed to actually evaluate it in a robust manner. We hope to demonstrate that the new methods are sufficiently general that they provide broad scope for tackling a variety of mathematical problems.

6 January 2012 **Yakov Sinai **(Princeton) *Ergodic Properties of Square-free Numbers*

Abstract: The talk is based on a joint paper with F.Cellarosi in which we show that the set of square-free numbers has a natural invariant measure with respect to which the shift is an ergodic automorphism with pure point spectrum.

13 January 2012 **Antti Knowles **(Harvard) *Spectral statistics of large random matrices *

Abstract: Large random matrices exhibit the striking phenomenon of universality: under very general assumptions on the matrix entries, the limiting spectral statistics coincide with those of a Gaussian matrix ensemble. I review recent results on the spectral universality of random matrices. I also describe two types of phase transition in random matrix models: one associated with heavy-tailed entries, and the other associated with finite-rank deformations. (Joint work with L. Erdos, H.T. Yau, and J. Yin.)

20 January 2012 **Jens Marklof** (Bristol) *From the Lorentz gas to random graphs: new applications of measure rigidity in statistical physics, number theory and combinatorics *

Abstract: Measure rigidity of flows on homogeneous spaces is a powerful tool that has recently seen many spectacular applications in number theory and mathematical physics. In this lecture I will discuss applications of measure rigidity to three seemingly unrelated problems: kinetic transport in the periodic Lorentz gas, diameters of random circulant graphs and Frobenius' coin exchange problem.

03 February 2012 **Gerard Besson** (Grenoble) *Natural maps: old and new*

Abstract : we intend to show, on very basic examples, how one can construct maps between closed (Riemannian) manifolds that have very nice properties regarding volume elements. We shall explain a couple of applications among which an elementary proof of Mostow rigidity for real hyperbolic manifolds and if time permits recent applications to growth of some groups. The talk is intended to be a colloquium talk.

10 February 2011 **Kevin Buzzard** (Imperial) *The p-adic numbers -- past, present and future*

Abstract: Many mathematicians are quite at ease doing calculations where they have a function *f(x*) which satisfies some equation, and in an attempt to get a feeling of what's going on they expand the function out as a power series *f(x*)=*a*+*bx*+*cx*^{2}+... and figure out the first few coefficients. Perhaps we know for some other reason that the answer is going to be a polynomial, and then this "expanding out" trick might even solve the problem completely. But if *f* isn't a polynomial then a mathematician is typically not too bothered, because it's just a power series, and power series work fine in most cases.

Number theorists sometimes try this trick, but instead of a function *f* they have a number *n*. Instead of* x* being a variable, it might be a "base" -- for example if *x*=2 and *a,b,c* ... are all 0 and 1, we're just expanding out a number in binary. So perhaps the analogue of "polynomial" is "base 2 expansion of a number". But whatever is the analogue of "power series" in this context? Can we really write 1+2+2^{2}+2^{3}+... and hope to be still doing meaningful mathematics?

The first half of the talk will be an explanation of why the answer is "yes we really can do this!". We will invent the 2-adic numbers, and more generally the *p-*adic numbers. The second half of the talk will be a survey of where these *p-*adic numbers show up within mathematics.

Both parts of the talk will be accessible to undergraduates. The first part might leave you with the impression that the *p-*adic numbers are just a quirky bit of fun -- but I will attempt to show in the second part that they are of essential importance in all sorts of attempts to resolve all sorts of hard problems (for example Wiles' proof of Fermat's Last Theorem used them crucially, and they also figure prominently in modern attacks on the Birch and Swinnerton-Dyer conjecture). Is there even a *p*-adic Langlands Philosophy? Why does google give 52000 hits for *p*-adic string theory? We will muse on these questions.

24 February 2012 **John Jones** (Warwick) *Exotic spheres and the Kervaire invariant*

Abstract: Exotic n-spheres are closed (compact with no boundary) differentiable manifolds which are homotopy equivalent to the standard n-sphere but are not diffeomorphic to it. The study of exotic n-spheres began with Milnor's construction of an exotic 7-sphere in 1956, and in 1963 Kervaire and Milnor effectively classified exotic n-spheres for n > 4 (in terms of the homotopy groups of spheres) modulo one very difficult problem which came to be known as the Kervaire invariant problem.

This problem was finally solved by Hill, Hopkins and Ravenel in 2009 and its solution is a major breakthrough in topology.

In this talk, which is intended for a general audience, I will try to explain what this Kervaire invariant problem is and also the impact its solution has in both differential topology and homotopy theory.

2 March 2012 **Gui-Qiang G. Chen** (Oxford) *Hyperbolic Conservation Laws and Divergence-Measure Fields *

Abstract: In this lecture we will start with an introduction of nonlinear conservation laws including their derivations, basic features, and connections with other areas such as fluid mechanics, relativity, differential geometry, and calculus of variations. Then we will discuss several mathematical challenges and fundamental issues we have to face for solving nonlinear hyperbolic conservation laws. The main focus will be on a class of weakly differentiable vector fields, called divergence-measure fields, and its natural connection with entropy solutions for hyperbolic conservation laws. In particular, we will discuss some recent efforts to establish a theory of divergence-measure fields toward developing analytical frameworks for studying entropy solutions of multidimensional hyperbolic conservation laws. Further connections, trends, and open problems in this direction will be also addressed.

9 March 2012 **Jean Bertoin **(Zurich) *Coagulation with limited aggregations*

Abstract: Smoluchowski's coagulation equations have been used as elementary mathematical models for the formation of polymers. We review here some recent contributions on a variation of this model in which the number of aggregations for each atom is a priori limited. Macroscopic results in the deterministic setting can be explained probabilistically at the microscopic level by considering a version of stochastic coalescence with limited aggregations, which can be related to the so-called random configuration model of random graph theory.

16 March 2012 **Christophe Ortner **(Warwick) *Continuum Elasticity from Molecular Mechanics*

Abstract: Multi-scale modelling has become a paradigm that transcends all scientific disciplines. A key challenge that arises in many scenarios is the connection between discrete (e.g., atomistic) and continuum descriptions of matter.

The Cauchy-Born rule postulates such a connection for crystal elasticity, which seems almost naive at first glance. Nevertheless, it has been found to provide an accurate description of crystal elasticity even at the smallest scales.

In this talk, I will describe various attempts to derive and understand the Cauchy--Born model and its limitations: Cauchy's derivation of linearized elasticity, the variational approach of the 90's, and most recently the point of view of approximation theory. Time permitting, I will mention some useful concepts to make the connection between molecular mechanics and Cauchy-Born elasticity rigorous.

27 April 2012 **Marc Timme** (Göttingen) *Unstable attractors, complex heteroclinic networks and computation by switching*

Abstract: Complex heteroclinic networks of dynamically connected saddle states persistently emerge in a broad range of high-dimensional systems. The dynamics close to heteroclinic networks is intrinsically robust, easily controllable, and provides a large number of state-changing options. It therefore seems a viable backbone for new forms of computation.

Here we introduce the concept of unstable attractors, periodic orbits that are linearly unstable but exhibit a basin of attraction of positive measure. Their stability structure induces switching processes between saddle orbits -- and heteroclinic networks of them -- in response to external perturbations. We analyze how symmetry-breaking inhomogeneities naturally induce predictable persistent switching dynamics across such networks. We show that such systems are capable of computing arbitrary logic operations by entering into switching sequences in a controlled way. This dynamics thus offers a highly flexible new kind of computation based on switching along complex heteroclinic networks of saddles.

04 May 2012 **Yuri Tomilov** (IM PAN Warsaw) *On rates in mean ergodic theorems*

The mean ergodic theorem in its simplest form states that the arithmetic means of powers of a power bounded operator on a reflexive Banach space converge in the strong operator topology. It is the starting point of many results in operator ergodic theory. In my talk, I am going to present a general framework for the study of rates of convergence in mean ergodic theorems. In particular, I will show how to unify and generalize various results dealing with rates in mean ergodic theorems in a number of cases.

11 May 2012 **Sam Payne** (Yale) *Nonarchimedean norms in algebraic geometry*

Abstract: The usual norm on the complex numbers and its associated analytic geometry (holomorphic functions and differential forms) have been fundamental tools for understanding the geometry and topology of complex algebraic varieties since the beginnings of the subject. Nonarchimedean norms, such as the p-adic norm on the rational numbers, also have an associated analytic geometry, which has been used in number theory, but is just beginning to be applied in other areas of mathematics, such as algebraic geometry and dynamical systems. This talk will be an introduction to nonarchimedean geometry with an explanation of its combinatorial manifestation in tropical geometry and relations to algebraic curves.

18 May 2012 **Benny Sudakov** (UCLA) *Induced Matchings, Arithmetic Progressions and Communication*

Abstract: Extremal Combinatorics is one of the central branches of discrete mathematics which deals with the problem of estimating the maximum possible size of a combinatorial structure which satisfies certain restrictions. Often, such problems have also applications to other areas including Theoretical Computer Science, Additive Number Theory and Information Theory. In this talk we will illustrate this fact by several closely related examples focusing on a recent work with Alon and Moitra.

25 May 2012 **Robert McKay** (Warwick) *Manufacturing Impact*

Abstract: Mathematics in the UK is facing increased demands to demonstrate impact, e.g. the REF requires impact case studies, EPSRC grant applications require Pathways to Impact plans, and EPSRC has commissioned a study into the Economic Impact of Mathematics Research.

The Mathematics research community is rising to the challenge, e.g. the IMA has produced a series of 20 "Mathematics Matters" case studies with EPSRC sponsorship and 6 more are to come with HE STEM sponsorship, the Council for Mathematical Sciences have been invited to put on a day of talks and discussion for the Parliamentary and Scientific Committee entitled "Mathematics Matters: a crucial contribution to the country's economy" (15 March) following publication of an article by Ken Brown and Paul Glendinning in "Science in Parliament", and Peter Rowlett coordinated a multi-author paper on "The unplanned impact of mathematics" in Nature (14 July 2011). Various members of this department have collaborations with industry, commerce, government, medicine and other domains.

EPSRC is expected to launch a call on "Mathematics and Manufacturing" shortly, which offers a superb opportunity for mathematics to have technological and economic impact. To give an idea of what is possible and stimulate ideas, I will recount my experience of a collaboration via Warwick Manufacturing Group with a robotics company called Metris UK, sponsored by the Technology Strategy Board. More opportunity will be available via a MIR@W day scheduled for 30 April.

15 June 2012 **Arnaud Chéritat** (Toulouse) *Open questions in holomorphic dynamics*

Abstract : after a computer assisted introduction to the topic of holomorphic dynamical systems in complex dimension one, I will present some of the main open questions of the field.