Past

Many portfolio optimization problems are directly or indirectly concerned with the current maximum of the underlying. For example, loockback or Russian options, optimization with max-drawdown constraint , or indirectly American Put Options, optimization with floor constraints.

The Azema-Yor martingales or max-martingales, introduced in 1979 to solve the Skohorod embedding problem, appear to be remarkably efficient to provide simple solution to some of these problems, written on semi-martingale with continuous running supremum.

The phenomenon of ignition is one with which we are all familiar, but which is remarkably difficult to define and model effectively. My own (description rather than definition) is “initiation of a (high temperature) self-sustaining exothermic process”; it may of course be desirable, as in your car’s engine, or highly undesirable, as the cause of many disastrous fires and explosions Both laboratory experiments and numerical simulations demonstrate its extreme sensitivity to external influences, past history and process (essentially chemical) kinetics, but at the heart of all instances there appears to be some “critical” unstable equilibrium state. Though some analytical modelling has been useful in particular cases, this remains in general virgin territory for applied mathematicians – perhaps there is room for some “knowledge transfer” here.

We recall that an elliptic curve is a curve of genus one with a rational point on it. Certain algorithms for determining the structure of the group of rational points on an elliptic curve produce a whole set of curves of genus one and then require that we determine which of these curves has a rational point.

Unfortunately no algorithm which has been proved to terminate is known for doing this. Such an algorithm or proof would probably have profound implications for the study of elliptic curves and may shed light on the Birch and Swinnerton-Dyer conjecture.

This talk will be about joint work with Samir Siksek (Warwick) on the development of a new algorithmic criterion for determining that a given curve of genus one has no rational points. Both the theory behind the criterion and recent attempts to make the criterion computationally practical, will be detailed.

Hilbert schemes classify subschemes of a given projective variety / scheme. They are special cases of Quot schemes which are moduli spaces for quotients of a fixed coherent sheaf. Hilb and Quot are among the first examples of moduli spaces in algebraic geometry, and they are crucial for solving many other moduli problems. I will try to give you a flavour of the subject by sketching the construction of Hilb and Quot and by discussing the role they play in applications, in particular moduli spaces of stable curves and moduli spaces of stable sheaves.

> There is currently tremendous interest in geometric PDE, due in part
> to the geometric flow program used successfully to attack the Poincare
> and Geometrization Conjectures.  Geometric PDE also play a primary
> role in general relativity, where the (constrained) Einstein evolution
> equations describe the propagation of gravitational waves generated by
> collisions of massive objects such as black holes.
> The need to validate this geometric PDE model of gravity has led to
> the recent construction of (very expensive) gravitational wave
> detectors, such as the NSF-funded LIGO project.  In this lecture, we
> consider the non-dynamical subset of the Einstein equations called the
> Einstein constraints; this coupled nonlinear elliptic system must be
> solved numerically to produce initial data for gravitational wave
> simulations, and to enforce the constraints during dynamical
> simulations, as needed for LIGO and other gravitational wave modeling efforts.
>
> The Einstein constraint equations have been studied intensively for
> half a century; our focus in this lecture is on a thirty-year-old open
> question involving existence of solutions to the constraint equations
> on space-like hyper-surfaces with arbitrarily prescribed mean
> extrinsic curvature.  All known existence results have involved
> assuming either constant (CMC) or nearly-constant (near-CMC) mean
> extrinsic curvature.
> After giving a survey of known CMC and near-CMC results through 2007,
> we outline a new topological fixed-point framework that is
> fundamentally free of both CMC and near-CMC conditions, resting on the
> construction of "global barriers" for the Hamiltonian constraint.  We
> then present such a barrier construction for case of closed manifolds
> with positive Yamabe metrics, giving the first known existence results
> for arbitrarily prescribed mean extrinsic curvature.  Our results are
> developed in the setting of a ``weak'' background metric, which
> requires building up a set of preliminary results on general Sobolev
> classes and elliptic operators on manifold with weak metrics. 
> However, this allows us to recover the recent ``rough'' CMC existence
> results of Choquet-Bruhat
> (2004) and of Maxwell (2004-2006) as two distinct limiting cases of
> our non-CMC results.  Our non-CMC results also extend to other cases
> such as compact manifolds with boundary.
>
> Time permitting, we also outline some new abstract approximation
> theory results using the weak solution theory framework for the
> constraints; an application of which gives a convergence proof for
> adaptive finite element methods applied to the Hamiltonian constraint.

We compute the high frequency limit of the Hemholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, we prove that the limiting Wigner measure satisfies a stationary transport equation with source term. This result encodes the refraction phenomenon. Second, we study the particular case when the index is constant in each media, for which the analysis goes further: we prove that the interface does not capture energy asymptotically in this case.

Euler equations for incompressible fludis describe the evolution of the divergence-free velocity of a non-viscous fluid (when viscosity is present, we have the well-known Navier-Stokes equations). V. Arnold discovered that they correspond to geodesic equations in the space of volume-preserving diffeomorphisms but several exemples show that it is not always possible to solve the corresponding variational problems inducing minimal energy displacements. A solvable relaxed version, in a non-deterministic setting (measures on the path space, with possible splitting of the particles), has been introduced by Y. Brenier who intensively studied the problem. Together with M. Bernot and A. Figalli we founded new solutions and characterization results. In the talk I'll present the most interesting features of the problem and of its solutions.

Mike Giles recently came up with a very general technique that improves the fundamental complexity of Monte Carlo simulation in the context where stochastic differential equations are simulated numerically. I will discuss some work with Mike Giles and Xuerong Mao that extends the theoretical support for this approach to the case of financial options without globally Lipschitz payoff functions. I will also suggest other application areas where this multi-level approach might prove valuable, including stochastic computation in cell biology.

New techniques in cell and molecular biology have produced huge advances in our understanding of signal transduction and cellular response in many systems, and this has led to better cell-level models for problems ranging from biofilm formation to embryonic development. However, many problems involve very large numbers of cells, and detailed cell-based descriptions are computationally prohibitive at present. Thus rational techniques for incorporating cell-level knowledge into macroscopic equations are needed for these problems. In this talk we discuss several examples that arise in the context of cell motility and pattern formation. We will discuss systems in which the micro-to-macro transition can be made more or less completely, and also describe other systems that will require new insights and techniques.

14.15 - 15.00 Part I

Marc Yor : The infinite horizon case.

15.00 - 15.15 A short break for questions and answers

15.15 - 16.00 Part II

Amel Bentata : The finite horizon case.

Roughly, the Black-Scholes formula is a distribution function of the maturity. This may be explained in terms of the last passage times at a given level of the underlying Brownian motion with drift.

Conversely, starting with last passage times up to finite horizon, we obtain a 2-parameter variant of the Black-Scholes formula.