Past

Virtually all decisions taken by living beings, from financial investments to life history, mate choice or anti-predator responses involve uncertainties and inter-temporal trade offs. Thus, hypothesis and formal models from these different fields often have heuristic value across disciplines. I will present theories and experiments about temporal discounting and risky choice originating in behavioural research on birds. Among other topics, I will address empirical observations showing risk aversion for gains and risk proneness for losses, exploring parallels and differences between Prospect Theory, Risk Sensitivity Theory and Scalar Utility Theory.

Motivated by the tensile experiments on titanium alloys of Petrinic et al

(2006), which show the formation of cracks through the formation and

coalescence of voids in ductile fracture, we consider the problem of

formulating a variational model in nonlinear elasticity compatible both

with cavitation and with the appearance of discontinuities across

two-dimensional surfaces. As in the model for cavitation of Müller and

Spector (1995) we address this problem, which is connected to the

sequential weak continuity of the determinant of the deformation gradient

in spaces of functions having low regularity, by means of adding an

appropriate surface energy term to the elastic energy. Based upon

considerations of invertibility we are led to an expression for the

surface energy that admits a physical and a geometrical interpretation,

and that allows for the formulation of a model with better analytical

properties. We obtain, in particular, important regularity properites of

the inverses of deformations, as well as the weak continuity of the

determinants and the existence of minimizers. We show further that the

creation of surface can be modelled by carefully analyzing the jump set of

the inverses, and we point out some connections between the analysis of

cavitation and fracture, the theory of SBV functions, and the theory of

cartesian currents of Giaquinta, Modica and Soucek. (Joint work with

Carlos Mora-Corral, Basque Center for Applied Mathematics).

We prove uniqueness of the Kerr black holes within the connected, non-degenerate, analytic class of regular vacuum black holes. (This is joint work with Piotr Chrusciel. arXiv:0806.0016)

A classical problem in differential geometry is to deform the metric of a given manifold so that some of its curvatures become prescribed functions. Classical examples are the Uniformization problem for compact surfaces and the Yamabe problem for compact manifolds of dimension greater than two.
We address a similar problem for the so-called Q-curvature, which plays an important role in conformal geometry and is a natural higher order analogue of the Gauss curvature. The problem is tackled using a variational and Morse theoretical approach.

This second 'problem sheet' of the term includes a proof of Jensen's Theorem for the number of zeroes of an analytic function in a disc, the usefulness of which is highlighted by an application to the Riemann zeta-function.

In this talk, we give the asymptotics estimates for the heat kernel and its gradient estimates on H-type groups. Moreover, we get gradient estimates for the heat semi-group.

Open Riemann surfaces and the Weil-Petersson Poisson structure

We discuss some connections between various notions of rationality in the face of uncertainty and the theory of convex risk measures, both in a static and a dynamic setting.

The circulatory system is the most important and amongst the most complicated mechanisms in the human body. Consisting of the heart, the arteries and the veins, it is amply aided by a variety of mechanisms aiming to facilitate adequate perfusion of the body tissues at the appropriate pressure. On this talk I am focusing on the development of a computational model which relates patient specific factors (age, gender, whether someone is an athlete/smokes etc) and their effects on different vascular regions which ultimately determine the arterial pressure and the cardiac output.

This paper provides a unifying approach for valuing contingent claims on a portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T; x)-bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T, and zero else. Necessary and sufficient conditions on the stochastic term structure movements for the absence of arbitrage are given. Background market risk as well as feedback contagion effects of the loss process are taken into account. Moreover, we show that any ex- ogenous specification of the volatility and contagion parameters actually yields a unique consistent loss process and thus an arbitrage-free family of (T; x)-bond prices. For the sake of analytical and computational efficiency we then develop a tractable class of doubly stochastic affine term structure models.

Sperm cells have been an archetype for very low Reynolds number swimming since the pioneering work of Gray & Hancock in the 1950s. However, there are fundamental questions regarding the swimming and function of mammalian, and particularly human sperm, that are unanswered, and moreover scientific and technological developments mean that for the first time, answering these questions is now possible.

I will present results of our interdisciplinary work on two topics: (1) the relatively famous problem of 'surface accumulation' of sperm, and (2) characterising the changes to the flagellar beat that occur in high viscosity liquids such as cervical mucus. The approach we use combines both mathematical modelling and high speed imaging experiments.

I will then discuss areas we are currently developing: quantifying the energy transport requirements of sperm, and understanding chemotaxis - the remarkable ability of human sperm to 'smell' lily of the valley perfume, which may be important in fertilisation.

This talk is concerned with the probabilistic behaviour of a condition

number C(A) for the problem of deciding whether a system of n

homogeneous linear inequalities in m unknowns has a non-zero solution.

In the case where the input system is feasible, the exact probability

distribution of the condition number for random inputs is determined,

and a sharp bound for the general case. In particular, for the

expected value of the logarithm log C(A), an upper bound of order

O(log m) (m the number of variables) is presented which does not

depend on the number of inequalities.

The probability distribution of the condition number C(A) is closely

related to the probability of covering the m-sphere with n spherical

caps of a given radius. As a corollary, we obtain bounds on the

probability of covering the sphere with random caps.

Modelling the behaviour of light in photonic crystal fibres requires

solving 2nd-order elliptic eigenvalue problems with discontinuous

coefficients. The eigenfunctions of these problems have limited

regularity. Therefore, the planewave expansion method would appear to

be an unusual choice of method for such problems. In this talk I

examine the convergence properties of the planewave expansion method as

well as demonstrate that smoothing the coefficients in the problem (to

get more regularity) introduces another error and this cancels any

benefit that smoothing may have.

To any Coxeter group (W,S) together with an appropriate representation on V one may associate various categories of "singular Soergel bimodules", which are certain bimodules over invariant subrings of

regular functions on V. I will discuss their definition, basic properties and explain how they categorify the associated Hecke algebras and their parabolic modules. I will also outline a motivation coming from geometry and (if time permits) an application in knot theory.

Let $(M,\omega)$ be a symplectic manifold, and $g$ a Riemannian metric on $M$ compatible with $\omega$. If $L$ is a compact Lagrangian submanifold of $(M,\omega)$, we can compute the volume Vol$(L)$ of $L$ using $g$. A Lagrangian $L$ is called {\it Hamiltonian stationary} if it is a stationary point of the volume functional amongst Lagrangians Hamiltonian isotopic to $L$.

Suppose $L'$ is a compact Lagrangian in ${\mathbb C}^n$ which is Hamiltonian stationary and {\it rigid}, that is, all infinitesimal Hamiltonian deformations of $L$ as a Hamiltonian stationary Lagrangian come from rigid motions of ${\mathbb C}^n$. An example of such $L'$ is the $n$-torus $ \bigl\{(z_1,\ldots,z_n)\in{\mathbb C}^n:\vert z_1\vert=a_1, \ldots,\vert z_n\vert=a_n\bigr\}$, for small $a_1,\ldots,a_n>0$.

I will explain a construction of Hamiltonian stationary Lagrangians in any compact symplectic manifold $(M,\omega)$, which works by `gluing in' $tL'$ near a point $p$ in $M$ for small $t>0$.

Random partial orders and random linear extensions

Several interesting models of random partial orders can be described via a

process that builds the partial order one step at a time, at each point

adding a new maximal element. This process therefore generates a linear

extension of the partial order in tandem with the partial order itself. A

natural condition to demand of such processes is that, if we condition on

the occurrence of some finite partial order after a given number of steps,

then each linear extension of that partial order is equally likely. This

condition is called "order-invariance".

The class of order-invariant processes includes processes generating a

random infinite partial order, as well as those that amount to taking a

random linear extension of a fixed infinite poset.

Our goal is to study order-invariant processes in general. In this talk, I

shall explain some of the problems that need to be resolved, and discuss

some of the combinatorial problems that arise.

(joint work with Malwina Luczak)

The relaxation of a free-energy functional which describes the

order-strain interaction in nematic liquid crystal elastomers is obtained

explicitly. We work in the regime of small strains (linearized

kinematics). Adopting the uniaxial order tensor theory or Frank

model to describe the liquid crystal order, we prove that the

minima of the relaxed functional exhibit an effective biaxial

microstructure, as in de Gennes tensor model. In particular, this

implies that the response of the material is soft even if the

order of the system is assumed to be fixed. The relaxed energy

density satisfies a solenoidal quasiconvexification formula.

Given K a separably closed field of finite ( > 1) degree of imperfection, and semiabelian variety A over K, we study the maximal divisible subgroup A^{sharp} of A(K). We show that the {\sharp} functor does not preserve exact sequences and also give an example where A^{\sharp} does not have relative Morley rank. (Joint work with F. Benoist and E. Bouscaren)

We present a theory for  stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points.
For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman  equation, in  the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. All  known examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We also prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem coincides with the equilibrium control and value function respectively for the time inconsistent problem. We also study some concrete examples.

Abstract available at: http://people.maths.ox.ac.uk/~kirby/LInnocente.pdf

We consider an infinite plate being withdrawn from an infinite pool of viscous liquid. Assuming that the effects of inertia and surface tension are weak, Derjaguin (1943) conjectured that the 'load', i.e. the thickness of the liquid film clinging to the plate, is determined by a certain formula involving the liquid's density and viscosity, the plate's velocity and inclination angle, and the acceleration due to gravity.

In the present work, Deryagin's formula is derived from the Stokes equations in the limit of small slope of the plate (without this assumption, the formula is invalid). It is shown that the problem has infinitely many steady solutions, all of which are stable - but only one of these corresponds to Derjaguin’s formula. This particular steady solution can only be singled out by matching it to a self-similar solution describing the non-steady part of the film between the pool and the film’s 'tip'. Even though the near-pool region where the steady state has been established expands with time, the upper, non-steady part of the film (with its thickness decreasing towards the tip) expands faster and, thus, occupies a larger portion of the plate. As a result, the mean thickness of the film is 1.5 times smaller than the load.

The results obtained are extended to order-one inclinantion angles and the case where surface tension is present.