Seminar on Microlocal Analysis and Applications

组织者 Organizer:Chen Xi(Fudan), Long Jin(Tsinghua)
时间 Time:10:00-11:00 am, July 29th(Fri), 2022
地点 Venue:Zoom: 618-038-6257, Password: SCMS

Upcoming talks:


Title: On The relative volume of Poincare-Einstein manifolds

Speaker: Wang Fang(SJTU)

Time: 10:00-11:00 am, July 29th (Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: For a Poincare-Einstein manifold, the Bishop-Gromov comparison theorem tells us that the relative volume is a non-increasing function of the geodesic radius. In this talk, I will show that the fractional Yamabe constant at the conformal infinity provides a lower bound for this function. As an application, this implies a gap phenomena and the rigidity theorem.

Past talks:


Title: Semiclassical analysis of elastic surface waves and inverse problems

Speaker: Zhai Jian(Fudan)

Time: 10:00-11:00 am, July 1st (Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: We present a semiclassical description of surface waves traveling in stratified elastic medium. This semiclassical perspective is known to the geophysical community as the slow variational principle, and its mathematically rigorous framework was proposed by Colin de Verdiere (for acoustics). I will also talk about related inverse spectral problems, that are to use the dispersion relations of surface waves to explore the subsurface structure.


Title: Inverse problems for non-linear wave equations

Speaker: Antonio Sa Barreto(Purdue)

Time: 10:00-11:00 am, June 17th (Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: Since the superposition principle does not hold for non-linear equations, their solutions interact and hence produce information that would not be present if the equation were linear. Of course the problem is how one can extract such information. I will discuss the inverse problem of determining a semi-linear potential in two different situations:

1) The inverse scattering problem for  Box u+ f(u)=0, when scattering holds for large initial data

2) Applications of nonlinear geometric optics to inverse problems. This can also be thought of as propagation of semiclassical singularities for non-linear wave equations.


Title: Scattering rigidity for analytic metrics

Speaker: Malo Jezequel(MIT)

Time: 10:00-11:00 am, June 3rd (Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: For analytic negatively curved compact connected Riemannian manifold with analytic strictly convex boundary, the scattering map for the geodesic flow determines the manifold up to isometry. After detailing this result, I will explain how it can be deduced from analytic wave front set computations involving a radial estimate in the analytic category. This is a joint work with Yannick Guedes Bonthonneau and Colin Guillarmou.



Title: Wave decay and non-decay in free space

Speaker: Kiril Datchev (Purdue)

Time: 10:00-11:00 am, May 20th (Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: We study the wave equation on geometric perturbations of Euclidean space, where the support of the perturbation is a compact set $K$. Local energy (i.e. energy over a bounded spatial region $U$) decays in time, in a way depending on the dynamics of the geodesic flow over $K$ and on the geometric relationship between $U$ and $K$. Simple examples show that $K$ can influence decay rates over $U$ even when the distance between $U$ and $K$ is large. For many radial problems it is possible to compute precisely the critical distance at which the influence stops and interpret this distance geometrically. In the general case the picture is less clear but some partial sharp results are known. Our approach to this problem is based on semiclassical resolvent estimates, proven in part using Olver's WKB approximations and in part using Carleman estimates. This talk is based on joint works with Long Jin and with Jeffrey Galkowski and Jacob Shapiro.


Title: Weyl laws for open quantum maps 

Speaker: Li Zhenhao (MIT)

Time: 10:00-11:00 am, May 6th (Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: Open quantum maps provide simple finite-dimensional models of open quantum chaos. They are families of $N \times N$ matrices that quantize a symplectic relation on a compact phase space, and their eigenvalues model resonances of certain open quantum systems in the semiclassical limit as $N \to \infty$. This makes them especially conducive to numerical experimentation and thus appealing in the study of scattering resonances. We consider a particular toy model that arises from quantizing the classical baker’s map. We find a Weyl upper bound in the semiclassical limit for the number of eigenvalues in a fixed annulus, and derive an explicit dependence on the radius of the annulus given Gevrey regularity. These results are accompanied by numerical experiments.


Title: Dynamics of resonances for 0th order pseudodifferential operators.

Speaker: Wang Jian(University of North Carolina, Chapel Hill)

Time: 10:00-11:00 am, Apr. 22th (Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: Zeroth order pseudodifferential operators on torus are studied as microlocal model of internal waves. These operators can have embedded eigenvalues. After 0th order analytic perturbations, the embedded eigenvalues become resonances and we prove a series expansion of the resonances. As results of the expansion, we obtain the Fermi golden rule for 0th order operators and we answer the question about the generic absence of embedded eigenvalues of 0th order operators asked by Colin de Verdiere.


Title: Semiclassical oscillating functions of isotropic type and their applications

Speaker: Wang Zuoqin(USTC)

Time: 10:00-11:00 am, Apr. 8th (Fri), 2022

Abstract: Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semiclassical analysis. In this talk I will describe how to associate classes of semiclassical oscillating functions to isotropic submanifolds of phase space, and show that these classes are invariant under the action of Fourier integral operators (modulo the usual clean intersection condition). Some sub-classes (coherent states, Hermite states) and applications will also be discussed. This is based on joint works with V. Guillemin (MIT) and A. Uribe (U. Michigan).



Title: Internal waves in 2D aquaria

Speaker: Maciej Zworski(University of California, Berkeley)

Time: 10:00-11:00 am, Mar. 25(Fri), 2022

Abstract: The connection between the formation of internal waves in fluids, spectral theory, and homeomorphisms of the circle was investigated by oceanographers in the 90s and resulted in novel experimental observations (Maas et al, 1997). The specific homeomorphism is given by a "chess billiard" and has been considered by many authors (John 1941, Arnold 1957... ). The relation between the nonlinear dynamics of this homeomorphism and linearized internal waves provides a striking example of classical/quantum correspondence (in a classical and surprising setting of fluids!). I will illustrate the results with numerical examples and explain how classical concepts such as rotation numbers of diffeomorphisms (introduced by Poincare) are related to solutions of the Poincare evolution problem. The talk is based on joint work with S Dyatlov and J Wang.

Title: Semiclassical analysis and the convergence of the finite element method

Speaker: Jared Wunsch(Northwestern University)

Time: 10:00-11:00 am, Mar. 11(Fri), 2022

Venue: Zoom: 618-038-6257, Password: SCMS

Abstract: An important problem in numerical analysis is the solution of the Helmholtz equation in exerior domains, in variable media; this models the scattering of time-harmonic waves. The Finite Element Method (FEM) is a flexible and powerful tool for obtaining numerical solutions, but difficulties are known to arise in obtaining convergence estimates for FEM that are uniform as the frequency of waves tends to infinity. I will describe some recent joint work with David Lafontaine and Euan Spence that yields new convergence results for the FEM which are uniform in the frequency parameter. The essential new tools come from semiclassical microlocal analysis. No knowledge of either FEM or semiclassical analysis will be assumed in the talk, however.