Past

> 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.

The task is to compute orthogonal eigenvectors (without Gram-Schmidt) of symmetric tridiagonals for isolated clusters of close eigenvalues. We review an "old" method, the Submatrix method, and describe an extension which significantly enlarges the scope to include several mini-clusters within the given cluster. An essential feature is to find the envelope of the associated invariant subspace.

In a financial market, the appreciate rates are very difficult to estimate precisely, and in general only some confidence interval will be estimated. This paper is devoted to the portfolio selection with the appreciation rates being in a certain closed convex set rather than some precise point. We study the problem in both expected utility framework and mean-variance framework, and robust solutions are given explicitly in both frameworks.

I will discuss some theorems of Chatzidakis, van den Dries, and Macintyre on definable sets over finite fields (Crelle 1992). This includes a geometric decomposition theorem for definable sets and a generalization of the Lang-Weil estimates, and uses model theory of finite and pseudo-finite fields.

If time permits, I shall mention a recent application of this work by Emmanuel Kowalski on new bounds for exponential sums (Israel Journal of Math 2007).

I would also like to mention some connections to the model theory of p-adic and motivic integrals and to general problems on counting and equidistribution of rational points.