Conference in Honor of Plamen Stefanov
Microlocal Analysis, Inverse Problems and Resonances
March Midwestern Microlocal Meeting
A Conference in Honor of Plamen Stefanov
Purdue University, West Lafayette, IN

March 22nd to 24th, 2019

Speakers



Schedule

Talks will take place on Friday, Saturday, and Sunday. They will be open to the mathematical public, but please register by emailing kdatchev@purdue.edu if you are planning to attend. There will also be a banquet on Saturday evening.

We will have rooms for speakers and participants at the Union Club Hotel on campus. This hotel has convenient parking and is connected to the Indianapolis and Chicago O'Hare airports by shuttles operated by Reindeer and Lafayette Limo.


Organizers
Kiril Datchev (Purdue), Antônio Sá Barreto (Purdue), Gunther Uhlmann (U Washington and HKUST), and Jared Wunsch (Northwestern).

Please email any questions to kdatchev@purdue.edu.


Funding

This meeting will be supported in part by Purdue University, the Simons Foundation, and the National Science Foundation.

Limited funding will be available for student and postdoc attendees. To apply, please submit a short cover letter and CV to kdatchev@purdue.edu, and ask your PhD advisor or another senior person familiar with your work to send a recommendation letter to the same address. Applications will be considered on an ongoing basis, but for full consideration please submit all materials by February 20th.


Abstracts

  • Lauri Oksanen

    Light ray transform and inverse problems for hyperbolic PDEs

    The problem to recover subleading terms in a wave equation given boundary traces of solutions to the equation can be reduced to the following problem in integral geometry: find a function (or a one form modulo a certain gauge invariance) given its light ray transfrom, that is, its integrals over all lightlike geodesics. It is an open question if the light ray transform is invertible even when the Lorentzian metric associated to the wave equation is close to the Minkowski metric. The Minkowski case was solved by Plamen Stefanov in 1989. We describe some recent results in product geometries, and discuss also a broken version of the light ray transform that arises when recovering the first order terms in a non-linear wave equation.


  • Gabriel Paternain

    Nonlinear detection of Hermitian connections in Minkowski space

    I will describe how to recover a Hermitian connection form the source-to-solution map of a cubic non-linear wave equation in Minkowski space; the equation is naturally motivated by the Yang-Mills-Higgs equations. The recovery is reduced to a geometric problem of independent interest and not considered before: recovering a connection from its broken non-abelian X-ray transform along light rays. This is joint work with Chen, Lassas and Oksanen.


  • Mikko Salo

    Fixed angle inverse scattering with two measurements

    We consider the fixed angle inverse scattering problem, and show that a compactly supported potential is uniquely determined by its scattering amplitude for two opposite fixed angles. We also show that a reflection symmetric potential is uniquely determined by its fixed angle scattering data. These results are proved by a reduction to a formally determined inverse problem for the wave equation in the time domain. We address several such problems for the wave equation, including the stable determination of a potential from boundary measurements related to two plane waves or to two waves corresponding to a point source and a spherical wave. These problems are of interest in geophysical applications. The proofs are based on reducing the inverse problem to a unique continuation type problem for the wave equation in the spirit of the Bukhgeim-Klibanov method, and on using a suitable Carleman estimate.

    This is joint work with Rakesh (Delaware).


  • John C. Schotland

    Quantum Optics in Random Media

    The theory of light-matter interactions in quantum optics is primarily concerned with systems consisting of a small number of atoms. We will review recent work on quantum optics in random media and show that in this setting, there is a close relation between the theory of spontaneous emission and kinetic equations for PDEs with random coefficients.


  • András Vasy

    Recovery of material parameters in transversally isotropic media

    In this talk I will discuss the recovery of material parameters in anisotropic elasticity, in the particular case of transversally isotropic media. I will indicate how the knowledge of the qSH (which I will explain!) wave travel times determines the tilt of the axis of isotropy as well as some of the elastic material parameters, and the knowledge of qP and qSV travel times conditionally determines a subset of the remaining parameters, in the sense that if some of the remaining parameters are known, the rest are determined, or if the remaining parameters satisfy a suitable relation, they are all determined, under certain non-degeneracy conditions. Furthermore, I will describe the additional issues, which are a subject of ongoing work, that need to be resolved for a full treatment.

    This is joint work with Maarten de Hoop and Gunther Uhlmann, and is in turn based on work with Plamen Stefanov and Gunther Uhlmann.