MIT PDE/Analysis Seminar, Fall 2004
At MIT in Building 2
In Room 2-142 at 4:00 PM on Wednesdays, unless otherwise noted.
February 11: Jared Wunsch (Northwestern)
``A Strichartz estimate for the Schrödinger equation on nontrapping
manifolds''
February 18: Frédéric Rochon (MIT)
``The determinant line bundle of cusp operators''
Abstract: If we consider a family of Dirac operators on a manifold with
boundary, then there is an associated family of Dirac operators defined on
the boundary. It is a relatively well-known result that the determinant
line bundle corresponding to this family must be trivial. In this talk, a
geometric proof of this result will be given using cusp pseudodifferential
operators. This is a joint work with Richard B. Melrose.
February 25: No seminar due to Simons lectures, part 3, by
Wendelin Werner (Paris Sud)
March 3:
March 10: Vitali Liskevich (University of Bristol)
``Heat kernel estimates for second-order elliptic operators with
lower order terms''
March 17:
March 24: Spring break
March 31: Sagun Chanillo (Rutgers)
``Symplectic Geometry and Recent Progress in the Treves Conjecture''
April 7: Dejan Slepcev (Toronto)
``Gradient-flow structure and stability of selfsimilar solutions of
thin-film and fast-diffusion equations''
Abstract: Nonlinear diffusion equations (porous-medium,
fast-diffusion equation) and thin-film
equations (with certain types of nonlinearities) can be recast as
gradient flows on a formal infinite-dimensional manifold.
The gradient-flow structure of these equations suggests a framework in
which to study stability of selfsimilar solutions.
In the case of porus-medium and fast-diffusion equations it provides a
way to show the asymptotic stability of the selfsimilar solutions, with
optimal rates of convergence.
Of particular interest to us will be the blow-up behaviour of
long-wave unstable thin-film equations and the stability of blow-up
profiles. Future directions and open problems will also be discussed.
April 14:
April 21:Karl-Theodor Sturm (Bonn)
``Mass transportation, entropy dissipation and nonlinear diffusions on
manifolds''
Abstract:
We discuss the problem of optimal mass transportation on a Riemannian
manifold, derive a differential inequality for the Jacobian of the
transport map, and deduce sharp criteria for convexity of energy
functionals (generalized entropies) on the space of probability measures
equipped with the Wasserstein distance.
Applications to nonlinear diffusions on manifolds will be presented.
Among others, this includes extensions to the nonlinear case of several
well known results on the trend to equilibrium, e.g.
the curvature-dimension condition of Bakry-Emery, Talagrand's inequality
and Gross' logarithmic Sobolev inequality.
In particular, the fast diffusion and porous medium equation will be
discussed.
April 28: James Ralston (UCLA)
``Isospectral Potentials''
May 5:
May 12: Maciej Zworski (UC Berkeley)
``Control theory and high energy eigenfuctions''