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• Tanya Christiansen
  
  Sharp constants and consequences for resonance counting for Schrödinger operators
  
  In a 2006 paper, Stefanov found the sharp constant in an upper bound on a resonance-counting function for Schrödinger operators on odd-dimensional Euclidean space (and for other operators). In this talk we explore work by several authors that builds on this paper. For example, the sharpness of the constant along with another estimate from that same paper allows us to show that (certain) potentials that have resonances regularly distributed in balls have their resonances regularly distributed in sectors as well. Dinh and Vu showed that the set of potentials supported in the ball of radius a and having resonances regularly distributed is dense in the set of all bounded potentials with support in this same ball. Borthwick and Borthwick–Crompton proved related results for Schrödinger operators on hyperbolic space.



• Venky Krishnan
  
  Momentum ray transforms
  
  Momentum ray transforms are certain weighted transforms that integrate symmetric tensor fields over lines in Eucildean space with weights that are powers of the integration parameter. This is a generalization of the standard longitudinal ray transforms which have attracted significant attention due to their many tomographic applications. We present an algorithm recovering the whole symmetric tensor field of rank m from its first m+1 momentum ray transforms. We then derive certain isometry relations between the tensor field and the momentum ray transforms, and use them to derive stability estimates.
  
  The talk is based on a joint work with Ramesh Manna (TIFR CAM, India), Suman Kumar Sahoo (TIFR CAM, India) and Vladimir Sharafutdinov (Sobolev Institute of Mathematics, Russia).



• Katya Krupchyk
  
  Bounds for Schrödinger operators with complex potentials and non-trapping metrics
  
  In this talk we shall discuss some recent progress on bounds of Keller and Lieb-Thirring type for eigenvalues of Schrödinger operators with complex potentials on non-trapping asymptotically conic manifolds. While in the self-adjoint case such bounds are classical, the non-self-adjoint case is considerably more involved and the sharpest results currently available were obtained quite recently by R. Frank and collaborators, in the Euclidean setting. We show that such results extend to the general setting of non-trapping asymptotically conic manifolds. A crucial ingredient in our proofs are weighted uniform estimates in suitable Schatten classes for the resolvent of the Laplacian on non-trapping asymptotically conic manifolds. This is joint work with Colin Guillarmou and Andrew Hassell.



• Ru-Yu Lai
  
  Parameter reconstruction for the transport equation
  
  The inverse problem for the transport equation finds applications in many areas such as optical imagine and semi-conductor designing. Roughly speaking it seeks to reconstruct certain physical properties of a medium from the data measured on the boundary. One widely studied model in optical imagine is the radiative transfer equation, which is a kinetic model for photon particles. In this talk I will discuss the determination of the optical parameters in the inverse transport problem and the one in the diffusion limit. This talk is based on a joint work with Qin Li and Gunther Uhlmann.



• François Monard
  
  Inversion of abelian and non-abelian ray transforms in the presence of statistical noise
  
  We will discuss two problems associated with ray transforms on simple surfaces:
  (1) how to reconstruct a function from its noisy geodesic X-ray transform (with applications to X-ray tomography)
  (2) how to reconstruct a skew-hermitian Higgs field from its noisy scattering data (with applications to Neutron Spin Tomography)
  
  For (1), the derivation of new mapping properties for the normal operator I*I allows to prove a Bernstein–von Mises theorem, about the statistical reliability of the Maximum A Posteriori (MAP) as a reconstruction candidate in a Bayesian statistical inversion framework, including a reliable assessment of the credible intervals. For (2), a non-linear problem whose injectivity for the noiseless case was established by Paternain–Salo–Uhlmann, the derivation of a new stability estimate allows one to prove contraction rates for certain Tikhonov regularizers. Numerical illustrations will be presented.
  
  Joint works with Gabriel Paternain and Richard Nickl (Cambridge).



• Carlos R. Montalto Cruz
  
  Photoacoustic imaging with variable sound speed
  
  In this talk we will present a summary of the collaborative work of Gunther Uhlmann and Plamen Stefanov in acoustic step of the problem of photoacoustic imaging. We will go over the ideas on their 2009 and 2011 work on uniqueness, stability and reconstruction of the electromagnetic source. We will explain how this inverse problem relates with observability estimates for the wave operator. Finally, we will see how to modify some of their ideas to take into account boundary interactions with the wave.



• 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.



• Vesselin Petkov
  
  Parametric resonances for the linear and nonlinear wave equation with time periodic potential
  
  It was proved that for some time periodic non-negative potentials q having compact support with respect to x some solutions of the Cauchy problem for the wave equation with potential q have exponentially increasing energy as the time goes to infinity. This phenomenon is called parametric resonance. We show that if one adds a nonlinear defocusing term then the solution of the nonlinear wave equation exists for all t and its energy is polynomially bounded as t goes to infinity for every choice of the periodic potential. We prove also that the high Sobolev norms of the solution of the nonlinear equation are still polynomially bonded as the time goes to infinity. Moreover, we prove that the zero solution of the nonlinear wave equation is instable if the corresponding linear equation has the property mentioned above.
  
  This is a joint work with N. Tzvetkov.



• Jianliang Qian
  
  Babich-like ansatz for point-source Maxwell's equations
  
  We propose a novel Babich-like ansatz consisting of an infinite series of dyadic coefficients (three-by-three matrices) and spherical Hankel functions for solving point-source Maxwell's equations in an inhomogeneous medium so as to produce the so-called dyadic Green's function. Using properties of spherical Hankel functions, we derive governing equations for the unknown asymptotics of the ansatz including the traveltime function and dyadic coefficients. By proposing matching conditions at the point source, we rigorously derive asymptotic behaviors of these geometrical-optics ingredients near the source so that their initial data at the source point are well-defined. To verify the feasibility of the proposed ansatz, we truncate the ansatz to keep only the first two terms, and we further develop partial-differential-equation based Eulerian approaches to compute the resulting asymptotic solutions. Numerical examples demonstrate that our new ansatz yields a uniform asymptotic solution in the region of space containing a point source but no other caustics.



• 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.



• Yang Yang
  
  Some inverse source and coefficient problems for the wave operator
  
  We will discuss two inverse problems for the acoustic wave equation and its generalization. The first is an inverse source problem where one attempts to determine an instantaneous source from the boundary Dirichlet data. We give conditions on unique and stable determination, and derive an explicit reconstruction formula for the source. The second is an inverse coefficient problem on a cylinder-like Lorentzian manifold (M, g) for the Lorentzian wave operator perturbed by a vector field A and a function q. We show that local knowledge of the Dirichlet-to-Neumann map (DN-map) stably determines the jets of (g, A, q) up to gauge transformations, and global knowledge of the DN-map stably determines the lens relation of g as well as the light ray transforms of A and q. This is based on joint work with Plamen Stefanov.



• Steve Zelditch
  
  Spectral asymptotics on stationary spacetimes
  
  Two cornerstones of spectral asymptotics on a Riemannian manifold are the Weyl law and the Gutzwiller(–Duistermaat–Guillemin) trace formula. They are basic results of non-relativistic quantum mechanics.
  
  My talk gives a relativistic generalization of these results to globally hyperbolic stationary spacetimes with a compact Cauchy hypersurface. Generalization to simple spacetimes with asymptotically flat Cauchy hypersurfaces is possible. Inverse problems are anticipated. Joint work with A. Strohmaier.



• Maciej Zworski
  
  Rough control for Schrödinger operators on 2-tori
  
  I will explain how the results of Bourgain, Burq and the speaker '13 can be used to obtain control and observability by rough functions and sets on 2-tori. We show that for the time dependent Schrödinger equation, any set of positive measure can be used for observability and controllability.
  
  For non-empty open sets this follows from the results of Haraux '89 and Jaffard '90, while for sufficiently long times and rational tori this can be deduced from the results of Jakobson '97.
  
  Other than tori (of any dimension; cf. Komornik '91, Anantharaman–Macia '14) the only compact manifolds for which observability holds for any non-empty open sets are hyperbolic surfaces. That follows from results of Bourgain–Dyatlov '16 and Dyatlov–Jin '17 and I will discuss the difficulty of passing to rougher sets in that case. Joint work with N Burq.