Beyond Part III

Young Researchers in Mathematics 2009

16-18 April 2009

Centre for Mathematical Sciences, Cambridge

The keynote talks (one per session) will be given by (talk times are provisional)



Talk Abstracts

Riemannian holonomy groups and calibrated geometry

Differential geometry is the study of manifolds -- multidimensional smooth spaces, often with nontrivial topology -- and of geometric structures upon them. One of the most basic kinds of geometric structure is a Riemannian metric, which tells you the lengths of smooth paths in the manifold. I will explain the definition of the holonomy group of a Riemannian metric, and Berger's classification of holonomy groups. The holonomy group measures what extra geometric structures are compatible with the Riemannian metric. So, Berger's classification is a list of the most interesting geometric structures in Riemannian geometry. They include Kahler manifolds, hyperkahler manifolds, Calabi-Yau manifolds, G2 and Spin(7) manifolds. All these are of interest in String Theory in theoretical physics.

In the last part of the talk, I will discuss calibrated submanifolds, which are special classes of submanifolds with minimal volume in Riemannian manifolds with special holonomy.

Saving lives with fluid mechanics

Mathematical models of fluid flows are employed to describe a wide range of phenomena and make predictions with practical consequences in many applications. In this presentation I will report on two distinct research projects, with life-saving consequences. This first concerns snow avalanches and the design of barriers to deflect, retard and arrest the oncoming flow. The second concerns the direct injection of therapeutic drugs into the brain to treat tumours and degenerative diseases.

Higher dimensional black holes

Black holes in more than four spacetime dimensions exhibit much richer behaviour than in four dimensions. The properties of higher-dimensional black holes will be reviewed and the implications for string theory discussed.

Counting, measure, and metrics

What is the "size" of a thing? The dimension of a vector space, the Euler characteristic of a topological space, the entropy of a probability space and the biodiversity of an ecosystem are all examples of "size". I will explain all of these and more. And although I will not give a general answer to the general question, I will expose some common patterns. I will focus in particular on a new notion of the "size" of a metric space, which springs directly from a categorical line of thinking and appears to encode some very interesting geometrical information.

Statistics: The Mathematics of Society

The field of Statistics has expanded hugely in scope and range of application from its original definition as pertaining to the condition of a state. Nevertheless, there is a continuing requirement for effective statistical models for social processes. I will show how Statistics provides a bridge between Mathematical Sciences and the Social Sciences, giving illustrations of the way in which Mathematics such as graph theory, hypergeometric functions and representation theory arises in the study of, for example, cohabitation, social survey data disclosure and migration.

Moduli spaces and wall-crossing

Moduli spaces play a central role in modern algebraic geometry. They are varieties whose points parametrize objects of some sort (e.g. modules or curves or vector bundles). Often, to get a nice moduli space, one has to first choose something called a stability condition, and then restrict attention to the corresponding set of stable objects. The resulting moduli space can change as one varies the stability condition, a phenomenon known as wall-crossing. This talk will be about wall-crossing and some tools that have recently been developed to help to understand it.

The solar tachocline: why is it there and what is it for?

In this talk I shall discuss the dynamics of the solar tachocline. This region at the base of the solar convection zone is believed to play a key role in the generation of the solar activity cycle and the attendant magnetic phenomena in the Sun. I shall review the observations and describe some of the key issues for maintenance of the tachocline.

Holography and string theory

I will discuss the idea of `holography' in gravity, which states that in a quantum theory of gravity the number of degrees of freedom scale like the area of the boundary of the system, not as the volume as one might naively expect. I will introduce the `AdS-CFT' correspondence, which is a recent concrete realization of holography, where a quantum gravity theory is conjectured to be physically equivalent to a lower dimensional field theory. Implications for quantum gravity will be described.

Hedging in large financial markets

After some preliminaries on financial mathematics, I will discuss how a large financial market can be modeled via a stochastic evolution equation in an infinite dimensional space. In the context of such a model, the issue of hedging can be studied in terms of a martingale representation theorem. The specific topic of hedging exotic equity derivatives with a portfolio of variance swaps will be treated as an example. This work is joint with Francois Berrier.

Bifurcation theory for localised states

Much of our understanding of nonlinear systems revolves around what the system 'settles down to' at long times, and how this changes qualitatively as we vary control parameters. The payoff for understanding the generic qualitative changes ('bifurcations') that occur is that often the same bifurcations appear in mathematical models and experiments with very different physics.
I will illustrate this idea by discussing the large collection of nonlinear dissipative systems (for example, from fluid mechanics) which, when subjected to uniform external forcing, produce (counterintuitively) spatially-localised responses. In the simplest case, the set of localised equilibrium states (and they're not solitons in the classical sense) has a nice structure and a convincing theoretical explanation, but it's also clear that this is only the tip of the iceberg.

Approximate structure

Lately people have been asking questions of the sort "How can one define an approximate X, to what extent does it resemble an actual X, and is this useful?" In my talk I will touch on such questions when X stands for "group", "polynomial" or "homomorphism". I will attempt to say what this has to do with finding prime numbers in particular patterns, for example arithmetic progressions such as 5,11,17,23,29.

From Klein Polynomials to Carbon-12

It is well-known that through stereographic projection, one can put a complex coordinate z on a spherical surface. Felix Klein studied the complex coordinates of the vertices, edge centres and face centres of each platonic solid this way, and found that they are the roots of rather simple polynomials in z. Related to these Klein polynomials there are some further, rational functions of z (ratios of polynomials), which have the same symmetries as the platonic solids.

Recently, it has been discovered that various model physical systems, in chemistry, condensed matter, nuclear and particle physics, have smooth structures with the same symmetries as platonic solids. The Klein polynomials and related rational functions are very useful for describing them mathematically.

The talk will end with a brief discussion of a model for atomic nuclei in which the protons and neutrons are regarded as close enough together to partially merge into a symmetric structure of this type, called a Skyrmion. Various small nuclei, up to carbon-12 and a bit larger, have been modelled this way.

Modern number theory as unifying factor for mathematics

Number theorists use in their research almost all areas of pure mathematics, and several fundamental developments in number theory serve as a unifying force in mathematics.
I will present several recent instances of fruitful interplay between number theory and algebra, geometry, topology, functional analysis and emerging new links to quantum physics.