## Mathematics Colloquium 2013 - 2014

### Programme

All seminars are held in Room E/0.15, Senghennydd Road, Cardiff at 15:10 unless stated otherwise.

Further information is available from Dr Timothy Logvinenko at LogvinenkoT@cardiff.ac.uk.

##### 13th November 2013

**Speaker: **Professor Sir Michael Berry FRS (H H Wills Physics Laboratory, University of Bristol, UK)

**Title:** The singularities of light: intensity, phase, polarization.

**Abstract: **Geometry dominates modern optics, in which we understand light through its singularities. These are different at different levels of description. At the coarsest level, where light is described in terms of geometrical-optics rays, are the singularities of bright light: caustics - focal lines and surfaces (the envelopes of ray families), classified by the mathematics of catastrophe theory. Wave optics smooths these singularities and decorates them with rich interference patterns, widely applicable, for example to rainbows, ship wakes and quantum scattering. Wave optics introduces a new quantity, namely phase, which has its own singularities. These are optical vortices: geometrically, the singularities of dark light - lines in space, or points in the plane. They occur in all types of quantum or classical waves. On a finer scale, where the vector nature of light cannot be ignored, the new phenomenon is polarization. This possesses its own singularities, also geometrical. As well as representing physics at each level with associated mathematics, these optical and wave geometries illustrate the idea of asymptotically emergent phenomena.

##### 4th December 2013

**Speaker:** Professor John King (University of Nottingham)

**Title:** Asymptotic behaviour in nonlinear diffusion.

**Abstract: **A variety of types of self-similarity can arise in classifying the asymptotic behaviour of parabolic partial differential equations, for example. A number of initial-value problems for a model nonlinear diffusion equation will be discussed, characterising in particular critical exponents and dimensions at which transitions occur.

##### 19th February 2014

**Speaker:** Professor Sir John Ball FRS (University of Oxford).

**Title:** Mathematics of interfaces in solids.

**Abstract: **Solid phase transformations give rise to a variety of unusual kinds of interfaces between different phases, some only observed in recent experiments. The lecture will discuss ways of describing and predicting these, and related questions of nonlinear analysis.

##### 5th March 2014

**Speaker:** Prof. Luc Jaulin (CNRS, Brest, France).

**Title:** Interval analysis for proving properties of dynamical systems; application to sailboat robotics.

**Abstract: **This talk presents a rigorous approach combining interval
analysis and Lyapunov theory for stability analysis of uncertain
dynamical systems.
The principle of the approach is to represent uncertain systems by
differential inclusions and then to perform a Lyapunov analysis in
order to cast the stability problem within a set-inversion framework.
With this approach, we can show that for all feasible perturbations,
(i) there exists a safe subset A of the state space the system cannot
escape as soon as it enters in it and (ii) if the system is outside A,
it cannot stay outside A forever. In a second step, the methodology is
used to build reliable robust controllers. An illustration related to
the line following problem of sailboat robots is then provided.
After introducing the basic notions on interval analysis and Lyapunov
theory, I will show how these tools can be used for stability analysis
of nonlinear systems. Then an experimental validation that took place
on January 2012 will then be presented. On this experiment, the
autonomous sailboat robot Vaimos, has gone from Brest to Douarnenez
(more than 100 km).

##### 9th April 2014

**Speaker:** Professor Nicola Garofalo (University of Padova)

**Title:** Recent developments in lower-dimensional obstacle problems.

**Abstract: **The study of the classical obstacle problem, initiated in the 60's with the pioneering works of G. Stampacchia, H. Lewy, J. L. Lions, has led to beautiful and deep developments in calculus of variations and geometric partial differential equations. The crowning achievement has been the development, due to L. Caffarelli, of the theory of free boundaries.

During the past decade there has been a great deal of activity on the lower dimensional obstacle problem, also known as Signorini problem. Such problem arises in a variety of situations of interest for the applied sciences: it presents itself in elasticity, when an elastic body is at rest, partially laying on a surface; in fluidodynamics it models the flow of a saline concentration through a semipermeable membrane when the flow occurs in a preferred direction; it also arises in financial mathematics in situations in which the random variation of an underlying asset changes discontinuously.

In this lecture I will overview what is currently known about the problem and I will discuss some new developments, such as for instance the time-dependent, or parabolic Signorini problem.

##### 11th June 2014

**Speaker:** Professor Sir Simon Donaldson FRS (Imperial College London)

**Title:** Invariant theory and Kahler metrics.

**Abstract: ** In algebraic geometry one is interested in constructing moduli spaces of various kinds, in particular of algebraic varieties up to equivalence. Many of the ideas go back to 19th century work on invariants, the simplest example being the cross-ratio of 4 points on the Riemann sphere. In complex differential geometry one is interested in the existence of various special metrics on an algebraic variety, for example any compact Riemann surface has a metric of constant Gauss curvature. These ideas come together: it is conjectured in general and proved in certain cases that the "stability" condition arising in the algebro-geometric moduli problem is just what is required for the existence of the appropriate metric. In the first part of the talk I will outline some of this general background. In the second part I will outline joint work with X.Chen and S. Sun which deals with Kahler Einstein metrics on Fano manifolds. There is an interesting interaction between Riemannian and complex geometry.

##### 8th October 2014

**Speaker:** Prof. Florin Boca (University of Illinois at Urbana-Champaign)

**Title:** TBC.

**Abstract: **TBC.

##### 5th November 2014

**Speaker:** Prof. Alison Etheridge (University of Oxford)

**Title:** Modelling evolution in a spatial continuum.

**Abstract: **TBC.