Oxford

A Hyperkähler manifold is a riemannian manifold carrying three complex structures which behave like quaternions such that the metric is Kähler with respect to each of them. This means in particular that the manifold is a symplectic manifold in many different ways. In analogy to the Marsden-Weinstein reduction on a symplectic manifold, there is also a quotient construction for group actions that preserve the Hyperkähler structure and admit a moment map. In fact most known (non-compact) examples of hyperkähler manifolds arise in this way from an appropriate group action on a quaternionic vector space.

In the first half of the talk I will give the definition of a hyperkähler manifold and explain the hyperkähler quotient construction. As an important application I will discuss the moduli space of solutions to the gauge-theoretic "Self-duality equations on a Riemann surface", the space of Higgs bundles, and explain how it can be viewed as a hyperkähler quotient in an infinite-dimensional setting.

The aim of this talk is to get a feel for the Ricci flow. The Ricci flow was introduced by Hamilton in 1982 and was later used by Perelman to prove the Poincaré conjecture. We will introduce the notions of Ricci flow and Ricci soliton, giving simple examples in low dimension. We will also discuss briefly other types of geometric flows one can consider.

Starting from a definition of the cohomology of a group, we will define the bounded cohomology of a group. We will then show how quasi-homomorphisms lead to cocycles in the second bounded cohomology group, and use this to look at the second bounded cohomology of some of our favourite groups. If time permits we will end with some applications.

I will introduce Symmetric spaces via a result of Kleiner & Leeb, comparing the axioms in their definition of a Euclidean building with properties of symmetric spaces of noncompact type.

Bourdon's building is a negatively curved 2-complex built out of hyperbolic right-angled polygons. Its automorphism group is large (uncountable) and remarkably rich. We study, and mostly answer, the question of when there is a discrete subgroup of the automorphism group such that the quotient is a closed surface of genus g. This involves some fun elementary combinatorics, but quickly leads to open questions in group theory and number theory. This is joint work with David Futer.

A behavioral mean-variance portfolio selection problem in continuous time is formulated and studied. Based on the standard mean-variance portfolio selection problem, the cumulative distribution function of the cash flow is distorted by a probability distortion function. Then the problem is no longer a convex optimization problem. This feature distinguishes it from the conventional linear-quadratic (LQ) problems.

The stochastic optimal LQ control theory no longer applies. We take the quantile function of the terminal cash flow as the decision variable.

The corresponding optimal terminal cash flow can be recovered by the optimal quantile function. Then the efficient strategy is the hedging strategy of the optimal terminal cash flow.