Symposium on Microlocal Analysis and Partial Differential Equations

Speaker: Rakesh, University of Delaware, USA

Abstract: Wave propagation in an inhomogeneous acoustic medium may be modeled, for example, by the wave operators □ + q, ρ∂²t − ∆ or ∂² t − ∆g, for a real smooth function q(x), a positive function ρ(x), or a Riemannian metric g(x) on Rⁿ, with q, ρ − 1, g − gEucl supported in a bounded domain. The medium is probed by plane waves coming from a finite number (dimension dependent) of directions, and the resultant time dependent waves are measured on the boundary of the domain. We describe our partial results, for the long standing open problems, about the recovery of q, ρ, g from these boundary measurements. These results were obtainedin collaboration with Lauri Oksanen and Mikko Salo.

Speaker: Uday Kiran, SSIHL, Puttaparathi, India

Abstract: In this talk, we discuss the microlocal analysis of a class of Fuchsian operators. These are pseudo-differential operators with real principal symbols whose characteristic varieties are the union of two smooth hypersurfaces intersecting non-involutively. We present various methods for determining their generalized inverses, wavefront set analysis and examine the branching of singularities phenomenon. The propagation of singularities provides insight into their solvability. Finally, we show that although higher-order Fuchsian operators have long been regarded as non-involutive, they nevertheless also exhibit propagation behavior similar to that of real principal type operators.

Speaker : Venky Krishnan, TIFR CAM, India

Abstract: We study the microlocal properties of the geodesic ray transform of symmetric m-tensor fields on two dimensional Riemannian manifolds with boundary allowing the possibility of conjugate points. As is known from an earlier work on the geodesic ray transform of functions in the presence of conjugate points, the normal operator can be decomposed into a sum of a pseudodifferential operator (PDO) and a finite number of Fourier integral operators (FIOs) under the assumption of no singular conjugate pairs along geodesics, which always holds in 2-dimensions. In this work, we use the method of stationary phase to explicitly compute the principal symbol of the PDO and each of the FIO components of the normal operator acting on symmetric m-tensor fields. Next, we construct a parametrix recovering the solenoidal component of the tensor fields modulo FIOs, and prove a cancellation of singularities result, similar to an earlier result of Monard, Stefanov and Uhlmann for the case of geodesic ray transform of functions in two dimensions. We point out that this type of cancellation result is only possible in the two dimensional case.

Speaker: Johannes Sjöstrand, Université Bourgogne, Dijon, France

Abstract: This talk is based on a joint work with Michael Hitrik and Martin Vogel. Let P(x, hDx; h) = p(x, hDx) + hp1(x, hDx) + h2p2(x, hDx) + ... be a differential operator in the semi-classical limit 0 < h ≪ 1. If the Poisson bracket

i ⁻¹ {p, p'}(x, ξ) = i⁻¹p′ξ· p′x − p′x· p′ξ(x, ξ)

does not vanish identically on the zero set of p, then P is non-self-adjoint and we know (cf. H ̈ormander 1960, ...) that P has singular values that are O(h^N ) ∀N ≥ 0, 0, and even O(exp −1/(Ch)) for some C > 0 in the analytic case. Also, P has approximate null solutions concentrated to points in phase space where p = 0, {p, p}/i > 0 and similarly P^∗ has approximate null solutions concentrated to points in phase space where p = 0, {p, p}/i < 0. One may ask for the precise exponential decay rate of the small singular values. We obtain such results for the ∂ operator on certain exponentially weighted L² spaces on a torus or on a cylinder, related to quantum tunneling between the regions in the energy surface p = 0 where {p, p}/i > 0 and {p, p}/i < 0, respectively.

Speaker : Luigi Rodino, Università di Torino, Italy

Abstract: The integration of operator kernels with the Wigner distribution, first conceptualized by E. Wigner in 1932 and later extended by L. Cohen and others, has opened new avenues in time-frequency analysis and operator calculus. Despite substantial advancements, the presence of “ghost frequencies” in Wigner kernels continues to pose significant challenges, particularly in the analysis of Fourier integral operators (FIOs) and their applications to partial differential equations (PDEs). In this talk, we build on the foundational concepts of Wigner analysis to introduce a novel framework for controlling ghost frequencies through combined Gaussian and Sobolev regularization techniques. By focusing on FIOs with non-quadratic phase functions, we develop rigorous estimates for the Wigner kernels that are crucial for their applicability to Schr ̈odinger equations with non-trivial symbol classes. Unlike previous approaches, our methodology not only mitigates the interference caused by ghost frequencies but also establishes robust bounds in the context of generalized symplectic mappings.

This is a joint work with Elena Cordero and Gianluca Giacchi. Ref: E. Cordero, G. Giacchi, and L. Rodino. Wigner analysis of operators. Part III: controlling ghost frequencies. arXiv:2412.01960.

Speaker : Gunther Uhlmann, University of Washington, Seattle, USA

Abstract : We describe some applications of microlocal analysis to inverse problems.