Location: Room 381T.
Time: Fridays 2:15-3:15pm.
Abstract: I will discuss semiclassical second microlocalization at a Lagrangian submanifold of T^*X, a precise way of measuring the failure of a distribution to be a Lagrangian distribution in the sense of Hörmander (translated into the semiclassical setting). One application is to the propagation of local Lagrangian regularity on invariant tori of systems with classically integrable hamiltonian.and
Abstract: We show that the scattering matrix at energy $n$ determines a conformally compact Einstein manifold modulo isometries. This is joint work with Colin Guillarmou.
Abstract: We will discuss a class of weakly differentiable vector fields, called divergence-measure fields. The primary focus is to introduce a suitable notion of the normal trace of any divergence-measure field over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. This requires a fundamental approximation theorem which states that, given a Radon measure that is absolutely continuous with respect to the (N-1)-dimensional Hausdorff measure, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the Radon measure. Then this approximation theorem is employed to derive the normal trace of any bounded divergence-measure field on the boundary of a set of finite perimeter as the limit of the normal traces of the field on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem holds on the set of finite perimeter. Furthermore, we will analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we will apply this framework to the derivation of systems of balance laws with measure-valued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. This talk will be mainly based on the recent joint work with M. Torres and W. Ziemer.
Abstract: We will discuss some recent results on Calderón's inverse problem of determining the electrical conductivity of a medium by making voltage and current measurements at the boundary. In particular we will consider the problem of determining the conductivity from partial data and the case of an anisotropic conductor.and
Abstract: The long-time asymptotics of this well-known model in chemotaxis cell movement will be analysed in the critical case. Its connection to free energies and the logarithmic HLS inequality will lead to the proof of infinite time aggregation in the critical case. A similar problem with nonlinear diffusion exhibiting analogous behavior will be studied in any dimensions. This corresponds to two works in collaboration with A. Blanchet and N. Masmoudi and A. Blanchet and P. Laurencot respectively.