Past

The space of smooth rational curves of degree d in projective space admits various moduli theoretic compactifications via GIT, stable maps, stable sheaves, Hilbert scheme and so on. I will discuss how these compactifications are related by explicit blow-ups and -downs for d

Consider the Erdos-Renyi random graph $G(n,p)$ inside the critical window, so that $p = n^{-1} + \lambda n^{-4/3}$ for some real \lambda. In

this regime, the largest components are of size $n^{2/3}$ and have finite surpluses (where the surplus of a component is the number of edges more than a tree that it has). Using a bijective correspondence between graphs and certain "marked random walks", we are able to give a (surprisingly simple) metric space description of the scaling limit of the ordered sequence of components, where edges in the original graph are re-scaled by $n^{-1/3}$. A limit component, given its size and surplus, is obtained by taking a continuum random tree (which is not a Brownian continuum random tree, but one whose distribution has been exponentially tilted) and making certain natural vertex identifications, which correspond to the surplus edges. This gives a metric space in which distances are calculated using paths in the original tree and the "shortcuts" induced by the vertex identifications. The limit of the whole critical random graph is then a collection of such

metric spaces. The convergence holds in a sufficiently strong sense (an appropriate version of the Gromov-Hausdorff distance) that we are able to deduce the convergence in distribution of the diameter of $G(n,p)$, re-scaled by $n^{-1/3}$, to a non-degenerate random variable, for $p$ in the critical window.

This is joint work (in progress!) with Louigi Addario-Berry (Universite de Montreal) and Nicolas Broutin (INRIA Rocquencourt).

I will review our current mathematical understanding of waves on black hole backgrounds, starting with the classical boundedness theorem of Kay and Wald on Schwarzschild space-time and ending with recent boundedness and decay theorems on a wider class of black hole space-times.

The talk will survey some recent and not so recent work on the

Hausdorff and box dimension of self-affine sets and related

attractors and repellers that arise in certain dynamical systems.

This talk will introduce Dirichlet's Theorem on the approximation of real numbers via rational numbers. Once this has been established, a stronger version of the result will be proved, viz Hurwitz's Theorem.

Random polymers are used to model various physical ( Ising inter- faces, wetting, etc.) and biological ( DNA denaturation, etc.) phenomena They are modeled as a one dimensional random walk (Xn), with excursion length distribution

P(E1 = n) = (n)=nc, c > 1, and (n) a slowly varying function. The polymer gets a random reward, whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (Vn). Depending on the relation be- tween the mean value of the disorder Vn and the temperature, the polymer might prefer to stick on the interface (pinning) or undergo a long excursion away from it (depinning).

In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the 'Pinning' model and also its relation to other directed polymer models

We study the class of Azema-Yor processes which are of the form F(M_t)-f(M_t)(X_t-M_t), where F'=f, X_t is a semimartingale with no positive jumps and M_t is its running maximum. We show that these processes arise as unique strong solutions to the Bachelier SDE which we also show is equivalent to the DrawDown SDE. The proofs are greatly simplified thanks to (algebraic) group property of the set of AY processes indexed by functions. We then restrict our attention to the case when X is a martingale. It turns out that the AY martingales are the only local martingales of the form H(X_t,M_t) for a Borel function H. Furthermore, they can also be characterised by their optimal

properties: all uniformly integrable martingales whose maximum dominates a given target are dominated by an AY martingale in the concave ordering of terminal values. We mention how these results find direct applications in portfolio optimisation/insurance theory.

Joint work with Laurent Cararro and Nicole El Karoui

Since summer 2007 financial markets moved in unprecedented ways. Volatility was extremely high. Correlations across the board increased dramatically. More importantly, also much deeper fundamental changes took place. In this talk we will concentrate on the following two aspects, namely, inter-bank unsecured lending at LIBOR and 40% recovery.

Before the crisis it was very realistic for the banks to consider that risk free rate of inter-bank lending, and hence also of funding, is equivalent to 3M LIBOR. This logic was extended to terms which are multiples of 3M via compounding and to arbitrary periods by interpolation and extrapolation. Driven by advances in financial mathematics arbitrage free term structure models have been developed for pricing of interest rate exotics, like LIBOR Market Model (or BGM). We explain how this methodology was challenged in the current market environment. We also point to mathematical questions that need to be addressed in order to incorporate in the pre-crisis pricing and risk management methodology the current market reality.

We also discuss historically validated and universally accepted pre-crisis assumption of 40% recovery. We expose its inconsistency with the prices observed now in the structured credit markets. We propose ways of addressing the problem and point to mathematical questions that need to be resolved.

We present a topos-theoretic interpretation of (a categorical generalization of) Fraïssé's construction in Model Theory, with applications to countably categorical theories. The proof of our main theorem represents an instance of exploiting the interplay of syntactic, semantic and geometric ideas in the foundations of Topos Theory.

Layered (or laminated) structures are increasingly used in modern industry (e.g., in microelectronics and aerospace engineering). Integrity of such structures is mainly determined by the quality of their interfaces: poor adhesion or delamination can lead to a catastrophic failure of the whole structure. Can nonlinear waves help us to detect such defects?

We study the dynamics of a nonlinear longitudinal bulk strain wave in a split, layered elastic bar, made of nonlinearly hyperelastic Murnaghan material. We consider a symmetric two-layered bar and assume that there is perfect interface for x 0, where the x-axis is directed along the bar. Using matched asymptotic multiple-scales expansions and the integrability theory of the KdV equation by the Inverse Scattering Transform, we examine scattering of solitary waves and show that the defect causes generation of more than one secondary solitary waves from a single incident soliton and, thus, can be used to detect the defect.

The theory is supported by experimental results. Experiments have been performed in the Ioffe Institute in St. Petersburg (Russia), using holographic interferometry and laser induced generation of an incident compression solitary wave in two- and three-layered polymethylmethacrylate (PMMA) bars, bonded using ethyl cyanoacrylate-based (CA) adhesive.

In this talk I will discuss how to construct generalized traces

and modified dimensions in certain categories of modules. As I will explain

there are several examples in representation theory where the usual trace

and dimension are zero, but these generalized traces and modified dimensions

are non-zero. Such examples include the representation theory of the Lie

algebra sl(2) over a field of positive characteristic and of Lie

superalgebras over the complex numbers. In these examples the modified

dimensions can be interpreted categorically and are closely related to some

basic notions involving the representation theory. This joint work with Jon

Kujawa and Bertrand Patureau.

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)

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.