Past

Seminar also with N. El Karoui and Y. Jiao

Dynamic modelling of default time for one single credit has been largely studied in the literature. For the pricing and hedging purpose, it is important to describe the price dynamics of credit derivative products. To this end, one needs to characterize martingales in the various filtrations and calculate conditional expectations by taking into account of default information, often modelized by a filtration $\bf{ D}$ generated by the jump process related to the default time $\tau$.

A general principle is to work with some reference filtration $\bf F$ which is often generated by some given processes. The calculations are then achieved by a formal passage between the enlarged filtration and the reference one on the set $\{\tau>t\}$ and the models are developed on the filtration $\bf F$.

In this paper, we are interested in what happens after a default occurs, i.e., on the set $\{\tau\leq t\}$. The motivation is to study the impact of a default event on the market, which will be important in a multi-credits setting. To this end, we adopt a new approach which is based on the knowledge of conditional survival probabilities. Inspired by the enlargement of filtration theory, we assume that the conditional law of $\tau$ admits a density.

We also present how our computations can be used in a multi-default setting.

In 1993 in his paper "A new strongly minimal set" Hrushovski produced a family of counter examples to a conjecture by Zilber. Each one of these counter examples carry a pregeometry. We answer a question by Hrushovski about comparing these pregeometries and their localization to finite sets. We first analyse the pregeometries arising from different variations of the construction before the collapse. Then we compare the pregeometries of the family of new strongly minimal structures obtained after the collapse.

Friction-driven vibration occurs in a number of contexts, from the violin string to brake squeal and machine tool vibration. A review of some key phenomena and approaches will be given, then the talk will focus on a particular aspect, the "twitchiness" of squeal and its relatives. It is notoriously difficult to get repeatable measurements of brake squeal, and this has been regarded as a problem for model testing and validation. But this twitchiness is better regarded as an essential feature of the phenomenon, to be addressed by any model with pretensions to predictive power. Recent work examining sensitivity of friction-excited vibration in a system with a single-point frictional contact will be described. This involves theoretical prediction of nominal instabilities and their sensitivity to parameter uncertainty, compared with the results of a large-scale experimental test in which several thousand squeal initiations were caught and analysed in a laboratory system. Mention will also be made of a new test rig, which attempts to fill a gap in knowledge of frictional material properties by measuring a parameter which occurs naturally in any linearised stability analysis, but which has never previously been measured.

The aim of this talk is to render the power of (short-step) interior-point methods for linear programming (and by extension, convex programming) intuitively understandable to those who have a basic training in numerical methods for dynamical systems solving. The connection between the two areas is made by interpreting line-search methods in a forward Euler framework, and by analysing the algorithmic complexity in terms of the stiffness of the vector field of search directions. Our analysis cannot replicate the best complexity bounds, but due to its weak assumptions it also applies to inexactly computed search directions and has explanatory power for a wide class of algorithms.

Co-Author: Coralia Cartis, Edinburgh University School of Mathematics.

It is well-known that Euclidean domains are PIDs; examples proving that the inclusion is strict are not commonly known. Here is one.

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