Analysis and PDE Seminar Homepage, Winter 2014

Location: 381U

Time: Monday, 4pm.

• Monday, January 27, Bay Area Microlocal Analysis Seminar at Stanford:

  • 3:15-4:15pm in 381U

    Gunther Uhlmann (University of Washington, visiting Stanford)

    ``Seeing Through Space Time''

    Abstract: We consider inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold. We formulate the concept of active measurements for relativistic models. We do this by coupling Einstein equations with equations for scalar fields. The inverse problem we study is the question, do the observations of the solutions of the coupled system in an open subset U of the space-time with the sources supported in U determine the properties of the metric in a larger domain? To study this problem we define the concept of light observation sets and show that these sets determine the conformal class of the metric. This corresponds to passive observations from a distant area of space which is filled by light sources. We will also consider inverse problems for other non-linear hyperbolic equations. This is joint work with Y. Kurylev and M. Lassas
  and
  • 4:30-5:30pm in 381U

    Maciej Zworski (University of California, Berkeley)

    ``Microlocal approach to dynamical zeta functions''

    Abstract: Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted “that a positive answer would be a little shocking”. Nevertheless the continuation was proved in 2012 by Giulietti--Liverani--Pollicott. In my talk I will present a proof of this result obtained by Dyatlov and myself and inspired by a trace formula of Guillemin and by recent work of Faure--Sjöstrand. It is based on a simple idea involving wave front sets and propagation of singularities: we apply methods of microlocal analysis to the generator of the flow, in particular, propagation of singularities results due to Duistermaat-Hörmander, Melrose and Vasy.

• Tuesday, January 28, 4pm, Room 383N (note unusual day/location):

  Richard Melrose (MIT)

  ``Analysis on loop spaces''

  Abstract: I will review the basic properties of loop spaces, the spaces of smooth maps from the circle into a manifold, and then emphasize some of the special properties they have as opposed to general Frechet manifolds. In particular I will describe the notion of `lithe' regularity and the role this plays in understanding the consequences of a manifold having a String structure. The results are joint work with Chris Kottke.

• Monday, February 3:

  Gunther Uhlmann (University of Washington, visiting Stanford)

  ``Travel Time Tomography, Boundary Rigidity, Lens Rigidity and Tensor Tomography''

  Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arise in several applications in geophysics and medical imaging among others.

  The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian metric with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

  We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary.

• Monday, February 10:

  Andras Vasy (Stanford)

  ``The local inverse problem for the geodesic X-ray transform and boundary rigidity within a conformal class (a.k.a. of sound speed)''

• Monday, March 3:

  Gunther Uhlmann (University of Washington, visiting Stanford)

  ``On Calderon's Inverse Problem''

  Abstract: We consider the inverse problem proposed by Calder\'on which consists on determining the electrical conductivity of a body by making voltage and current measurements at the boundary. Mathematical this consists on determining the coefficient of a second order elliptic partial differential equation from its associated Dirichlet-to-Neumann map. We concentrate on the case of partial measurements where the measurements are made on an open subset of the boundary.

• Monday, March 10:

  Kazuo Akutagawa (Tokyo Institute of Technology)

  ``The Yamabe problem on edge-cone manifolds and Aubin's inequality''

  Abstract.

• Monday, March 17, Bay Area Microlocal Analysis Seminar at Berkeley, in Evans Hall Room 891:

  • 2-3pm

    Michael Christ (Berkeley)

    ``On an inverse problem concerning Bergman kernels''

  and
  • 3:30-4:30pm

    Austin Ford (Stanford)

    ``The wave trace on manifolds with conic singularities''

    Abstract: We consider the trace of the (half-)wave group on a compact manifold with conic singularities. The trace of the wave group, which on the one hand equals $\sum e^{-it\lambda_j}$ where $\lambda_j^2$ are the eigenvalues of the Laplacian, is on the other hand a distribution in $t$ which is singular at the lengths of closed geodesics. Those closed geodesics that interact with the cone points generically do so "diffractively", carrying singularities into regions of phase space inaccessible to ordinary geodesic flow. We describe a formula for the leading order singularity of the wave trace at the lengths of closed diffractive geodesics, generalizing the formula due of Duistermaat and Guillemin in the smooth setting and that of Hillairet in the setting of flat surfaces with conic singularities.

• Schedule for Autumn 2013.
• Schedule for Spring 2013.
• Schedule for Winter 2013.
• Schedule for Autumn 2012.
• Schedule for Winter/Spring 2012.
• Schedule for Autumn/Winter 2010-2011.
• Schedule for Winter-Spring 2010.
• Schedule for Autumn 2009.
• Schedule for Spring 2009.
• Schedule for Winter 2009.
• Schedule for Spring 2008.
• Schedule for Winter 2008.
• Schedule for Autumn 2007.