## Mathematics Colloquium 2011 - 2012

### Programme

All seminars are held in room M/2.06, Senghennydd Road, Cardiff at 15:10 unless stated otherwise.

Further information is available from Dr Federica Dragoni at DragoniF@cf.ac.uk

##### 19th October 2011

**Speaker: **Mihalis Dafermos

**Title:** The black hole stability problem in general relativity.

**Abstract: **I will review recent progress on the mathematical

study of the black hole stability problem in general relativity.

##### 2nd November 2011

**Speaker: **Holger Dette (Bochum).

**Title:** Optimal designs, orthogonal polynomials and random matrices.

**Abstract: **The talk explains several relations between different areas of mathematics: Mathematical statistics, random matrices and special functions. We give a careful introduction in the theory of optimal designs, which are used to improve the accuracy of statistical inference without
performing additional experiments. It is demonstrated that for certain regression models orthogonal polynomials play an important role
in the construction of optimal designs. In the next step these results are connected with some classical facts from random matrix theory.

In the third part of this talk we discusss some new results on special functions and random matrices. In particular we analyse random bandz matrices, which generalize the classical Gaußschen ensemble. We show that the random eigenvalues of such matrices behave similarly as the deterministic roots of matrix orthogonal polynomials with varying recurrence coefficients. We study the asymptotic zero distribution of such polynomials and demonstrate that these results can be used to find the asymptotic proporties of the spectrum of random band matrices.

##### 16th November 2011

**Speaker: **Stephane Nonnenmacher

**Title:** Various aspects of "open quantum chaos".

**Abstract:**By "quantum chaos" one usually denotes the study of quantum (or wave)
systems which, in the semiclassical limit, converge to chaotic
dynamical systems. Yet, this "semiclassical convergence" is rather
singular, so that extracting from it precise information on the long time
quantum dynamics (e.g. informations on the spectrum of the quantum
dynamics) is not obvious. Here we will focus on two types of "open" chaotic systems. On one
side, we will consider scattering systems, where most trajectories escape
to infinity, but there also exist trajectories trapped for ever; when
the dynamics of the latter is chaotic, one speaks of "chaotic
scattering". These systems include for instance hyperbolic surfaces of infinite
volume, with connections in group or number theory. Our main objective
is to understand how the spectrum of quantum resonances is related to
the classical dynamics, in particular the structure of the trapped set.

I will describe a different "open" chaotic system, namely scalar waves propagating on a compact manifold of negative curvature, in presence of damping. The generator of such a system is nonselfadjoint, leading to complex eigenvalues (instead of resonances). The structure of the spectrum (in particular the presence of a spectral gap) allows to quantitatively estimate the time decay of the wave energy.

The methods of analysis for both types of systems are quite similar, they are based on the phenomenon of hyperbolic dispersion induced by the classical instability.

##### 14th December 2011 at 15:10 in Room E/0.15.

**Speaker: **Professon Kevin Glazebrook (Lancaster)

**Title:** Stochastic scheduling: A short history of index policies and some key recent developments.

**Abstract:** A multi-armed bandit problem concerns N≥2 independent populations of rewards whose statistical properties are unknown (or at least only partly known). A decision-maker secures rewards by sampling sequentially from the populations, using past sampled values to make inferences about the populations and so guide the choice of which population to sample next. The goal is to make these choices in such a way as to maximise some measure of total reward secured. Such problems embody in a particularly simple form the dichotomy present in many decision problems between making decisions with a view to securing information which can improve future decision-making (exploration) and those which exploit the information already available (exploitation). In the 1970s John Gittins discovered that important classes of such multi-armed bandit problems have solutions of a particularly simple form: at each stage of the sampling compute an index (the Gittins index) for each of the N populations, namely a function of the rewards already sampled from the population concerned. Always sample next from the population with the largest index. Moreover, the index concerned has a simple interpretation as an equivalent known reward for the population concerned.

It emerges that many problems involving the sequential allocation of effort, some of quite different character to the above multi-armed bandit problems, have index solutions. Since the 1970s, Gittins’ index result together with a range of developments and reformulations of it have constituted an influential stream of ideas and results contributing to research into the scheduling of stochastic objects. Application areas to which these ideas have contributed include approximate dynamic programming, the control of queuing systems , fast fashion retail, machine maintenance, military logistics, optimal search, research planning, sensor management, communication channel usage and website morphing. The talk will give an overview of some key ideas and some recent developments.

##### 15th February 2012

**Speaker: **Professor Heath-Brown FRS (Oxford)

**Title:** Zeros of systems of quadratic forms.

**Abstract:**Given a field k how large must the integer n be to ensure that every
quadratic form over k, in more than n variables, has a non-trivial zero
over k? What happens when we have r quadratic forms?

The talk will discuss these questions for fields k of arithmetic interest, particularly the p-adic fields.

##### 14th March 2012 in Room M/0.40.

**Speaker: **Professor Nick Higham

**Title:** Recent Progress in Matrix Functions.

**Abstract:**Functions of matrices are widely used in science, engineering and the
social sciences, due to the succinct and insightful way they allow
problems to be formulated and solutions to be expressed. New
applications involving matrix functions are regularly being found,
ranging from small but difficult problems in medicine to huge, sparse
systems arising in the solution of partial differential equations.

I will outline the underlying theory, algorithms and available software,
focusing on recent developments and pointing out promising avenues for

further research.