Cambridge

The theory of risk measurement has been extensively developed over the past ten years or so, but there has been comparatively little effort devoted to using this theory to inform portfolio choice. One theme of this paper is to study how an investor in a conventional log-Brownian market would invest to optimize expected utility of terminal wealth, when subjected to a bound on his risk, as measured by a coherent law-invariant risk measure. Results of Kusuoka lead to remarkably complete expressions for the solution to this problem.

The second theme of the paper is to discuss how one would actually manage (not just measure) risk. We study a principal/agent problem, where the principal is required to satisfy some risk constraint. The principal proposes a compensation package to the agent, who then optimises selfishly ignoring the risk constraint. The principal can pick a compensation package that induces the agent to select the principal's optimal choice.

I will start by reviewing the current status of the stability

problem for black holes in general relativity. In the second part of the

talk I will focus on a particular (symmetry) class of five-dimensional

dynamical black holes recently introduced by Bizon et al as a model to

study gravitational collapse in vacuum. In this context I state a recent

result establishing the asymptotic stability of the five dimensional

Schwarzschild metric with respect to vacuum perturbations in the given

class.

We call a family of permutations A in Sn 'intersecting' if any two permutations in A agree in at least one position. Deza and Frankl observed that an intersecting family of permutations has size at most (n-1)!; Cameron and Ku proved that equality is attained only by families of the form {σ in Sn: σ(i)=j} for i, j in [n].

We will sketch a proof of the following `stability' result: an intersecting family of permutations which has size at least (1-1/e + o(1))(n-1)! must be contained in {σ in Sn: σ(i)=j} for some i,j in [n]. This proves a conjecture of Cameron and Ku.

In order to tackle this we first use some representation theory and an eigenvalue argument to prove a conjecture of Leader concerning cross-intersecting families of permutations: if n >= 4 and A,B is a pair of cross-intersecting families in Sn, then |A||B|

Cover a plane with curves, one curve through each point

in each direction. How can you tell whether these curves are

the geodesics of some metric?

This problem gives rise to a certain closed system of partial

differential equations and hence to obstructions to finding such a

metric. It has been an open problem for at least 80 years. Surprisingly

it is harder in two dimensions than in higher dimensions. I shall present

a solution obtained jointly with Robert Bryant and Mike Eastwood.

We consider one-dimensional Brownian motion conditioned (in a suitable

sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.5860... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). I will also describe other conditionings of Brownian motion in which this principle of entropic repulsion manifests itself.

Joint work with Itai Benjamini.