# Mathematics Colloquium 2012-13 Abstracts

• 05 October 2012 Dwight Barkley (Warwick) Lecturing maths electronically - what I learned from Geometry and Motion
Abstract:
In the future we can expect undergraduate mathematics, especially at the first-year level, to be taught very differently from what we typically do today. Last year I delivered the first-year module Geometry and Motion by computer. I will present what I did, what was right and what was wrong, and then discuss ideas for further developments.
• 12 October 2012 Barbara Fantechi (SISSA, Trieste) Stacky viewpoint on Intersection theory
Abstract:
We briefly review the basic features of Fulton-MacPherson Intersection Theory, focusing on the key definition of Gysin pullback via degeneration to the normal cone. We then show how introducing stacky language leads to a slight generalization of the morphisms for which Gysin pullback is defined and a natural introduction to virtual pullbacks.
• 19 October 2012 David Evans (Cardiff) The search for the exotic - subfactors and conformal field theories
Abstract:
Subfactor theory provides a framework for studying modular invariant partition functions in conformal field theory, and candidates for exotic modular tensor categories. I will start with the background from the basics of subfactor theory. I will then describe some recent work which is also motivated by links with twisted equivariant K-theory through the K-theoretic realisation of Freed-Hopkins-Teleman of the Verlinde algebra of primary fields. This is joint work with Terry Gannon.
• 26 October 2012 Tom Sanders (Oxford) Approximate Groups
Abstract:
Our aim is to discuss the structure of subsets of Abelian groups which behave 'a bit like' cosets (of subgroups). One version of 'a bit like' can be arrived at by relaxing the usual characterisation of cosets: a subset S of an Abelian group is a coset if for every three elements x, y, z ∈ S we have x + y - zS. What happens if this is not true 100% of the time but is true, say, 1% of the time? It turns out that this is a situation which comes up quite a lot, and we shall discuss one possible answer called Freĭman’s theorem
• 02 November 2012 Cornelia Drutu (Oxford) On limit spaces and actions of groups
Abstract:
Using tools from logic one can construct the limit of a sequence of spaces, or of a sequence of actions of a group. Such limits are then used in topology (compactifications of spaces of representations, for instance of the Teichmueller space), geometry (Gromov's Polynomial Growth Theorem, isoperimetric inequalities, rigidity results) and analysis (local theory of Banach spaces, Assouad's theorem on embeddings, amenability in the sense of von Neumann, fixed point properties). The range of distinct limits for a given infinite group relates to the Continuum Hypothesis. I shall explain the construction of the limit and several of the applications mentioned above.
• 09 November 2012 Oliver Jenkinson (Queen Mary, London) Go forth and multiply by 10: digit expansions and optimal inequalities
Abstract:
Every rational number has an eventually periodic decimal expansion. The periodic part determines a finite orbit, invariant under multiplication by 10 (modulo 1). Fixing the arithmetic mean of the orbit, we consider questions such as: What is the smallest possible variance around the mean? Which orbit has largest geometric mean?
• 16 November 2012 Roger Heath-Brown (Oxford) Diophantine equations, Algebra, Geometry, Analysis & Logic
Abstract:
Diophantine equations are polynomial equations in two or more variables, for which one seeks integer, or rational, solutions. This talk will be an elementary introduction to the theory, showing how methods from algebra, geometry, analysis and logic are all of relevance.
• 23 November 2012 Athanasios Tzavaras (University of Crete) On the structure and properties of the equations of nonlinear elasticity
Abstract:
The equations of elastodynamics are a paradigm of a system of conservation laws where the lack of uniform convexity of the stored energy function poses challenges in the mathematical theory. Nevertheless, the existence of certain nonlinear transport equations for null-Lagrangeans reinforces the efficacy of the entropy as a stabilizing factor and recovers the strength associated wth uniformly convex entropies in hyperbolic systems. It turns out that elastodynamics with polyconvex stored energy can be embedded into a larger symmetric hyperbolic system and visualized as constrained evolution leading to a variational approximation scheme and an existence theory for measure valued solutions satisfying certain kinematic constraints in the weak sense. It provides a framework, in conjunction with the relative entropy method, to establish convergence of viscocity approximations or convergence of time-step approximants to smooth solutions of polyconvex elastodynamics. In addition, when a smooth solution is present it is unique within the class of measure valued solutions.
• 30 November 2012 Jon Keating (Bristol) Primes, Polynomials and Random Matrices
Abstract:
The Prime Number Theorem tells us roughly how many primes lie in a given long interval. We have much less knowledge of how many primes lie in short intervals, and this is the subject of a deep conjecture due to Goldston and Montgomery. Likewise, we also have much less knowledge of how many primes lie in different arithmetic progressions. This is the subject of an important conjecture due to Hooley. I will discuss the analogues of these conjectures for polynomials defined over function fields and explain how they can be proved using the theory of random matrices.
• 07 December 2012 - Meeting in Lieu of Colloquium Caroline Series (Warwick) Department Open Meeting: What is Athena Swan all about?
This week's event is not a colloquium but an open departmental meeting, chaired by Caroline Series, to explain the Athena Swan process. Everyone including students at all levels is welcome. Athena Swan is a national initiative which aims to tackle the unequal representation of women in science. It makes awards to STEM departments in Universities based on submission of a lot of structured evidence showing that gender equality and other diversity issues are being taken seriously. The Department has set up a working group with the aim of putting in an application for a Bronze level award next year.
The meeting will start with an introduction to the Athena Swan process and then be open for general discussion. This is an opportunity for everyone to contribute their thoughts and ideas on the subject, which will feed into our submission and action plan. If you are unable to come to the meeting and have comments or ideas, please feel free to discuss with either Caroline or Nav.
For further information about Athena Swan see http://www.athenaswan.org.uk/ and http://www2.warwick.ac.uk/services/equalops/athena
• 11 January 2013 Eric Smith (Santa Fe) Generating functional methods for non-equilibrium Markov processes

Abstract:
Equilibrium thermodynamics is the large-deviations theory of distributions whose entropies depend only on time-reversal-invariant state variables. The principles by which equilibrium thermodynamic descriptions are constructed extend readily to non-equilibrium systems if ensembles of states are replaced with ensembles of histories, though the resulting entropies are different from the equilibrium forms and many calculations become technically more difficult. The many-particle collective effects which are described by large-deviations theory are often the dynamically important and robust degrees of freedom in mediating multiscale dynamics in systems of interest in biology or hydrodynamics. In this talk I discuss practically useful methods to approximate the non-equilibrium generating functional that generalizes the equilibrium free energy, and the effective action which is its Legendre dual and the large- deviation function for configurations, and which generalizes the equilibrium fluctuation in entropy. I discuss properties of the Hamiltonian dynamical systems which arise by construction and characterize the problem of inferring history-dependent fluctuations for Markov processes, and demonstrate some results that parallel results that have been of interest in quantum field theory.

• 25 January 2013 Simon Ruijsenaars (Leeds) Calogero-Moser systems: A crossroads in mathematics and physics
Abstract:
The Calogero-Moser systems are integrable N-particle systems that are connected to a great many subfields of pure and applied mathematics, and that also find applications in various areas of physics. In this lecture we aim to survey this class of systems and their manifold relations to other subjects. As an illustration, we sketch the connection to the solitons of the classical and quantum versions of the sine-Gordon equation u_{tt} - u_{xx} = sin (u).
• 01 February 2013 Ben Green (Cambridge) On the Sylvester-Gallai theorem

Abstract:
The Sylvester-Gallai Theorem states that, given any set P of n points in the plane not all on one line, there is at least one line through precisely two points of P. Such a line is called an ordinary line. How many ordinary lines must there be? The Sylvester-Gallai Theorem says that there must be at least one but, in recent joint work with T. Tao, we have shown that there must be at least n/2 if n is even and at least 3n/4 - C if n is odd, provided that n is sufficiently large. These results are sharp. The talk will give an overview of this problem and the work towards its solution.

• 08 February 2013 Sylvia Serfaty (Paris 6 and Courant Institute) Coulomb gas, Renormalized energy and Abrikosov lattice

Abstract:
In superconductivity, one observes in certain regimes the emergence of densely packed point vortices forming perfect hexagonal lattice patterns. These are named Abrikosov lattices in physics. In joint work with Etienne Sandier, we showed how the distribution of these vortices is governed by a Coulomb type of interaction, which can be computed via a "Coulombian renormalized energy" which we introduced and derived rigorously from the Ginzburg-Landau model of superconductivity. Such an interaction turns out to be common in two-dimensional systems. We showed it arises in particular in the statistical mechanics of the "Coulomb gas", which contains as a speci^Lfic case the Ginibre ensemble of random matrices. We also defi^Lned a one-dimensional log-interaction analogue, arising naturally in the statistical mechanics of "log gases", which contains as a speci^Lc case the so-called GUE ensemble of random matrices. In this talk I will present the renormalized energy, examine the ques- tion of its minimization and its link with the Abrikosov lattice and weighted Fekete points. I will describe its relation with the statistical mechanics models mentioned above and show how it leads to expecting crystallisation in the low temperature limit.

• 15 February 2013 Alan Sokal (New York University and University College London) Roots $x_k(y)$ of a formal power series $f(x,y) = \displaystyle \sum\limits_{n=0}^\infty a_n(y) \, x^n$, with applications to graph enumeration and $q$-series

• 22 February 2013 Marc Lackenby (Oxford) A polynomial upper bound on Reidemeister moves
Abstract:
Consider a diagram of the unknot with c crossings. There is a sequence of Reidemeister moves taking this to the trivial diagram. But how many moves are required? In my talk, I will give an overview of my recent proof that there is there is an upper bound on the number of moves, which is a polynomial function of c.

• 01 March 2013 Rich Schwartz (Brown University) Thomson's 5-electron problem
Abstract:
In 1904, Thomson asked how N electrons would arrange themselves on the sphere so as to minimize their electrostatic potential. For the cases N=2,3,4,6,12, the electrons arrange themselves in the obvious maximally symmetric way. The case N=5 remained open for quite some time. I'll sketch my computer assisted, but rigorous, proof that the unique energy minimizer for the case N=5 is the triangular bi-pyramid.

• 08 March 2013 Roger Plymen (Southampton) 2 by 2 matrices as stepping-stones
Abstract:
I would like to show how, using the special linear group SL(2) as a stepping-stone, one can land, fairly quickly, on some serious and diverse parts of mathematics.
Part 1 -- from SL(2,C) to relativistic wave equations
Part 2 -- from SL(2,R) to Plancherel measure and Connes-Kasparov
Part 3 -- from SL(2,Q_p) to the local Langlands conjecture.

• 26 April 2013 David Blecher (Houston) Noncommutative topology and prescribing behaviour of noncommutative functions on noncommutative subsets
Abstract:
This will be a nontechnical survey on recent work on noncommutative topology and noncommutative Urysohn lemmas (finding noncommutative versions of functions (usually from a fixed algebra of operators) that have certain behaviours on certain noncommutative sets. In classical peak interpolation the setting is a subalgebra A of C(K), the continuous scalar functions on a compact Hausdorff space K, and one tries to build functions in A which have prescribed values or behaviour on a fixed closed subset E of K (or on several disjoint subsets). The sets E that 'work' for this are the p-sets, namely the closed sets whose characteristic functions are in the 'second annihilator' (or weak* closure) of A. Glicksberg's peak set theorem characterizes these sets as the intersections of peak sets, i.e. sets F for which there is a norm 1 function f in A which is 1 exactly on F. The typical peak interpolation result, originating in results of E. Bishop, says that if f is a strictly positive function in C(K), then the continuous functions on E which are restrictions of functions in A, and which are dominated by the 'control function' f on E, have extensions h in A with |h(x)| dominated by f(x) on all of K. It also yields 'Urysohn type lemmas' in which we find functions in A which are 1 on E and close to zero on a closed set disjoint from E. We discuss our generalizations of these results where A above is replaced by an algebra of operators on a Hilbert space, and open and closed sets are replaced by Akemann's noncommutative topology, which we explain. In fact we have recently been able to essentially complete this theory.

This is mostly joint work, with Damon Hay, Matt Neal, and Charles Read.

• 03 May 2013 Paul Krapivsky (Boston) Zero-Temperature Dynamics of Ising Ferromagnets
Abstract
I will discuss the fate of Ising ferromagnets endowed with a zero-temperature non-conservative dynamics, both the microscopic spin-flip dynamics and the macroscopic dynamics based on the time-dependent Ginzburg-Landau equation. After reviewing the situation in three dimensions, I will present evidence of a deep connection between the zero-temperature coarsening of both the two-dimensional time-dependent Ginzburg-Landau equation and the kinetic Ising model with critical continuum percolation. This connection allows to predict the probabilities of reaching a variety of topologically distinct metastable stripe states as these probabilities turn out to be related with crossing probabilities from critical percolation.

• 10 May 2013 Nicholas J. Higham (Manchester) Accuracy and Reproducibility of Numerical Computations
We discuss a number of aspects of accuracy and reproducibility of results computed by numerical algorithms in finite precision arithmetic, including:

• How to reformulate expressions to allow more accurate evaluation, including by use of the unwinding number.
• The need for, and exploitation of, higher precision (e.g., quadruple precision) arithmetic.
• Why repeating a (deterministic) numerical computation may lead to a different result and the implications of this behaviour.
• Abnormally small relative errors and their effects on performance profiles.

The talk will be accessible to those who are not specialists in numerical analysis.

• 10 May 2013 Mark Jerrum (Queen Mary, London) Phase transitions and computational tractability
"Phase transition" is a term that formally applies to infinite systems. But the effects of a phase transition can be felt in computations on finite
problem instances. It is widely appreciated that phase transitions are a barrier to the effective application of certain algorithmic techniques such as Markov chain Monte Carlo. But in fact one can sometimes exploit the existence of a phase transition to rule out an efficient approximation algorithm of any kind. Occasionally, the point at which a phase transition occurs can be rigorously linked to the exact boundary between tractability and intractability for a computational problem.

• 17 May 2013 Tullio Ceccherini-Silberstein (Universita' del Sannio Benevento) Cellular automata, amenable groups, surjunctivity, and sofic groups
Abstract:
The purpose of this talk is to overview the theory of cellular automata on groups focusing on the Garden of Eden Theorem for cellular automata over amenable groups and Gottschalk's surjunctivity problem. We shall present a few examples and discuss the notions of amenability and soficity and, possibly, the relation between surjunctivity and Kaplansky's conjecture on stable finiteness of group rings. The talk, completely self-contained, addresses to a wide audience, especially graduate students, interested in dynamical systems, group theory and ring theory, and theoretical computer science.

• 24 May 2013 Damiano Testa (Warwick) Rational points on surfaces

Abstract:

Diophantine problems typically ask for the solutions in integers or rational numbers to polynomial equations with integral coefficients:

the case of Pythagorean triples is one of the earliest instances of such a problem, and concerns the integral solutions to the equation x^2+y^2=z^2. For systems of polynomial equations whose complex solutions define a curve the situation is well-understood: the Mordell Conjecture, proved by Faltings, shows that the genus of the curve, a coarse geometric invariant, determines whether the set of rational points can be infinite or must be finite. In general, the expectation is that geometric properties of the set of complex solutions to polynomial equations should have a similar influence on the set of rational points of the same system. For the higher dimensional case though, the evidence is quite scarce and comes mostly from extensive computer searches.

In this talk, I will focus on the case of surfaces and I will present different examples that should conjecturally exhibit some of the possible behaviours.

• 31 May 2013 Philip Welch (Bristol) Turing Unbound: black holes, computers, neural nets, bangs, shrieks, transfinite ordinals, and the kitchen sink
Abstract:
There has been much talk (some of it rather loose) in the last few years concerning the possibility of "hypercomputation". This can start out from discussions as to whether Turing's machine model is appropriate for "modern" computation, or from claims that we can - perhaps only theoretically - compute "beyond the Turing limit", that is somehow fix up a device that enables us to 'compute' the Turing halting problem. On closer inspection most of such devices are seen to be formally cheating in some way or other, usually by assuming infinite precision measurement of some kind. We look at the mathematics of one model that at least is not cheating in that sense: these are computations in Malament-Hogarth spacetimes. We investigate as a purely logical exercise the limits of computation in such manifolds.

• 14 June 2013 Ian Leary (Southampton) Platonic polygonal complexes
Abstract:
Platonic polygonal complexes are a generalization of the platonic solids, the regular tesselations of the Euclidean and hyperbolic planes and the two-dimensional part of higher-dimensional regular tesselations such as the walls of the regular tesselation of 3-space by cubes. I shall discuss joint work with Tadeusz Januszkiewicz, Raciel Valle and Roger Vogeler on classifying them.

• 21 June 2013 Jeremy Quastel (Toronto) The Kardar-Parisi-Zhang equation and universality class
Abstract:
The KPZ equation is a stochastic partial differential equation used to describe randomly evolving interfaces. It is a member of a large universality class characterized by unusual size and distribution of fluctuations. Among the surprising developments in the last few years has been the discovery of a number of exact formulas. The talk will describe these, along with the background.