Seminar on Microlocal Analysis and Applications

Speaker:Minghui Ma (Chinese Academy of Sciences)
Organizer:Chen Xi(Fudan), Long Jin(Tsinghua)
Time:10:00-11:00 am, Mar. 31(Fri), 2023
Venue:Zoom: 618-038-6257, Password: SCMS

Upcoming talks: 

Title: Semiclassical analysis, geometric representation and quantum ergodicity  

Speaker:  Minghui Ma (Chinese Academy of Sciences)

Time: 10:00-11:00 am, Mar. 31(Fri), 2023

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


Quantum Ergodicity (QE) is a classical topic in spectral geometry, which states that on a compact Riemannian manifold whose geodesic flow is ergodic with respect to the Liouville measure, the Laplacian has a density one subsequence of eigenfunctions that tends to be equidistributed. In this talk, we present the QE for unitary flat bundles. By using a mixture of semiclassical and geometric quantizations, we can deal with the high frequency eigensections of a series of unitary flat bundles simultaneously. We will also give some applications on flat sphere bundles over hyperbolic surfaces.

Past talks:

Title: Resonant forms at zero for dissipative Anosov flow

Speaker:  Gabriel Paternain (University of Cambridge)

Time:  8:00-9:00 pm, Mar. 17(Fri), 2023

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


The Ruelle zeta function is a natural function associated with the periods of closed orbits of an Anosov flow, and it is known to have a meromorphic extension to the whole complex plane. The order of vanishing of the Ruelle zeta function at zero is expected to carry interesting topological and dynamical information and can be computed in terms of certain resonant spaces of differential forms for the action of the Lie derivative on suitable spaces with anisotropic regularity. In this talk I will explain how to compute these resonant spaces for any transitive Anosov flow in 3D, with particular emphasis in the dissipative case, that is, when the flow does not preserve any absolutely continuous measure. A prototype example is given by the geodesic flow of an affine connection with torsion and we shall see that for such a flow the order of vanishing drops by 1 in relation to the usual geodesic flow due to the Sinai-Ruelle-Bowen measure having non-zero winding cycle. This is joint work with Mihajlo Cekić.

Title: Microlocal Analysis and Inverse Problems

Speaker:  Gunther Uhlmann (University of Washington & HKUST)

Time: 10:00-11:00 am, March 3rd(Fri), 2023

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


We will discuss some applications of microlocal analysis to inverse problems, in particular the back scattering problem, Calderon's problem and inverse problems for nonlinear equations.

Title: Long time dynamics of Yang-Mills-Higgs equations and applications

Speaker: Yang Shiwu (Peking University)

Time: 10:00-11:00 am, December 30th(Fri), 2022

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


The Yang-Mills-Higgs eqautions are the nonabelian generalization of Maxwell-Klein-Gordon system, which models the motion of charged particles in electromagnetic field. It is well known that such system admits global smooth solutions in general globally hyperbolic spacetimes. In this talk, we will show that the solutions  in the future of a hyperboloid asymptotically decay like linear solutions for data bounded in some weighted energy space. The proof relies on vector field method and conformal invariance structure of the system. As applications, we also discuss the backward scattering problem with radiation data on the future null infinity.

Title: Propagation of Singularities on Kerr-de Sitter Spacetimes

Speaker:  Qiuye Jia (Stanford University)

Time: 10:00-11:00 am, December 16th(Fri), 2022

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


Kerr-de Sitter spacetimes is a family of slowly rotating black holes with positive cosmological constant. They are parameterized by their mass and angular momentum. We discuss propagation of singularities of wave equations on Kerr-de Sitter spacetimes. We construct a new pseudodifferential operator algebra using blow up and prove a propagation estimate improving the result of Hintz. The key property we use is the normally hyperbolic trapping and the special structure of the dual metric function of Kerr called `compensable' in this talk.

Title: Spectral asymptotics for kinetic Brownian motion on Riemannian manifolds    
Speaker: Tao Zhongkai(University of California, Berkeley)

Time: 10:00-11:00 am, Novermber 18th(Fri), 2022

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

Abstract: Kinetic Brownian motion is a stochastic process that interpolates between the geodesic flow and Laplacian. It is also an analogue of Bismut’s hypoelliptic Laplacian. We prove the strong convergence of the spectrum of kinetic Brownian motion to the spectrum of base Laplacian for all compact Riemannian manifolds. This generalizes recent work of Kolb--Weich--Wolf on constant curvature surfaces. As an application, we prove the optimal convergence rate of kinetic Brownian motion to equilibrium (given by the spectral gap of base Laplacian) conjectured by Baudoin--Tardif. This is based on joint work with Qiuyu Ren.

Title: A new approach to the nonlinear Schrödinger equation

Speaker:  Andrew Hassell (Australian National University)

Time: 10:00-11:00 am, Novermber 4th(Fri), 2022

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

Abstract: With collaborators Jesse Gell-Redman and Sean Gomes, we have begun to set up an entirely new framework for tackling the linear and nonlinear Schrödinger equation. I will describe this setup and explain why I believe it is a more powerful framework than existing approaches for studying nonlinear scattering and soliton dynamics.


Title: Fredholm approach to the Schrödinger equation

Speaker: Jesse Gell-Redman (University of Melbourne)

Time: 10:00-11:00 am, October 21st(Fri), 2022

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

Abstract: We discuss a new approach, inspired by work of Hintz and Vasy, to solving the Schrödinger equation $(i \partial_t - \Delta) u = f$ using the Fredholm method. Specifically, we use 'parabolic' pseudodifferential operators (reflecting the parabolic nature of the symbol of $P = i \partial_t - \Delta$) to obtain families of function spaces $X, Y$ for which $P : X \to Y$ is an isomorphism. The spaces further allow us to read off precise regularity and decay information about $u$ directly from that of $f$. We discuss applications to the nonlinear Schrödinger equation, and extensions of this method to equations with compact spatial perturbations, such as smooth decaying potential functions, using the N-body calculus of Vasy. This includes joint work with Dean Baskin, Sean Gomes, and Andrew Hassell.


Title: The Feynman propagator in some model singular settings

Speaker: Dean Baskin(Texas A&M University)

Time: 10:00-11:00 am, October 7th(Fri), 2022

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

Abstract: In this talk I will describe the existence and asymptotic properties of the Feynman propagator in three model singular settings: the scalar wave equation on cones, the scalar wave equation on Minkowski space with an inverse square potential, and the massless Dirac equation in 3 dimensions coupled to a Coulomb potential. The proof combines techniques of Gell-Redman–Haber–Vasy as well as prior work with Booth, Gell-Redman, Marzuola, Vasy, and Wunsch. One novelty of the proof is that it does not rely on Wick rotation (though a shadow of it survives in some special function analysis at infinity).


Title: Scattering theory for models in Conformal Field Theory

Speaker: Colin Guillarmou(Laboratoire de Mathematiques d'Orsay, Universite Paris-Saclay)

Time: 8:00-9:00 pm, October 7th(Fri), 2022

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

Abstract:  I will explain a famous model of 2 dimensional Conformal Field Theory called the Liouville CFT and discuss several aspects related to it, including the scattering analysis of its Hamiltonian. This is based on joint work with Kupiainen, Rhodes and Vargas.


Title:Non-line-of-sight imgaing

Speaker: Qiu Lingyun(Tsinghua University)

Time: 10:00-11:00 am, September 9th(Fri), 2022

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

Abstract: Non-line-of-sight imaging aims at recovering obscured objects from multiple-scattered light. It has recently received widespread attention due to its potential applications, such as autonomous driving, rescue operations, and remote sensing. However, in cases with high measurement noise, obtaining high-quality reconstructions remains a challenging task. In this work, we establish a unified regularization framework, which can be tailored for different scenarios, including indoor and outdoor scenes with substantial background noise under both confocal and non-confocal settings. The proposed regularization framework incorporates sparseness and non-local self-similarity of the hidden objects as well as smoothness of the measured signals. We show that the estimated signals, albedo, and surface normal of the hidden objects can be reconstructed robustly even with high measurement noise under the proposed framework. Reconstruction results on synthetic and experimental data show that our approach recovers the hidden objects faithfully and outperforms state-of-the-art reconstruction algorithms in terms of both quantitative criteria and visual quality.



Title: A two term Kuznecov sum formula

Speaker: Xi Yakun(Zhejiang University)

Time: 10:00-11:00 am, August 26th (Fri), 2022

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

Abstract: A period integral is the average of a Laplace eigenfunction over a compact submanifold. Much like for the Weyl law, one can obtain improved estimates on period integrals given geometric or dynamical assumptions on the geodesic flow. While there are many results improving bounds on period integrals, there have been none which improve the remainder of the corresponding sum formula. In this talk, we discuss a recent joint work with Emmett Wyman. We show that an improvement to the remainder term of this sum formula reveals a lower-order oscillating term whose behavior can be described by the dynamics of the geodesic flow. Moreover, this oscillating second term illuminates bounds on period integrals.



Title: Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures

Speaker: Liu Bochen(Southern University of Science and Technology)

Time: 10:00-11:00 am, August 12th (Fri), 2022

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

Abstract: We obtain new results on mixed-norm estimates of orthogonal projections. In the proof we interpolate analytically, not only on $p,q$, but also on dimensions of measures. We also introduce a new quantity called $s$-amplitude, to present our results and illustrate our ideas. This mechanism provides new perspectives on operators with measures, thus has its own interest.


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.




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.