Seminar on Microlocal Analysis and Applications

Speaker:Junshan Lin (Auburn University)
Organizer:Chen Xi(Fudan), Jin Long (Tsinghua)
Time:11:00-12:00 am, May 17 (Fri), 2024
Venue:Zoom: 618-038-6257, Password: SCMS

Upcoming talks: 

Title: A Mathematical Perspective on Scattering Resonances Through Sub-wavelength Holes
Speaker: Junshan Lin (Auburn University)

Time: 11:00-12:00 am, May 17 (Fri), 2024

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


The so-called extraordinary optical transmission (EOT) through metallic nanoholes has triggered extensive research in modern plasmonics and their applications in bio-sensing, imaging, etc. In this talk, I will present quantitative mathematical theory to understand scattering resonances that induce the EOT phenomenon in a series of metallic structures with sub-wavelength holes.



Past talks: 


Title: Holonomy Inverse Problem
Speaker: Mihajlo Cekić (Auburn University)

Time: 8:00-9:00 0m, May 2 (Thu), 2024

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


A choice of a connection (on a vector bundle) enables one to parallely transport vectors along curves. Holonomy inverse problem poses the following question: can one uniquely determine a connection (up to natural equivalence) from parallel transport (holonomy) along a suitably chosen set of geodesic curves? On closed (compact and boundaryless) Riemannian manifolds, under a dynamical hypothesis ensuring that the geodesic flow is chaotic, I will demonstrate in many situations that only traces of holonomy along closed geodesics suffice to determine the connection. Moreover, I will show it is possible to complement the uniqueness result with a suitable stability estimate of Hölder type, i.e. that the traces of holonomy determine stably the connection. By recent embedding results, this has consequences for the analogous problem on manifolds with boundary. The proofs are based on a combination of microlocal analysis, analysis of resonances, and hyperbolic dynamics. This is joint work with Thibault Lefeuvre.

Title: Inverse scattering on non-compact manifolds with general metric
Speaker: Hiroshi Isozaki(University of Tsukuba)
Time: 10:00-11:00 am, Apr 19(Fri), 2024

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


We consider a non-compact Riemannian manifold $M$ having ends equipped with asymptotically warped product metric. We allow any topology and metric for the finite part of $M$. As for the volume growth of each end, we assume any polynomial or exponential order. Studying the spectral properties of the Laplacian of $M$, we show that the S-matrix determines the manifold $M$. We can also include polynomially or exponentially shrinking cusp ends. This is a joint work with Matti Lassas in the university of Helsinki.

Title: Semiclassical asymptotics for Bergman projections
Speaker: Xiong Haoren(UCLA)
Time: 10:00-11:00 am, Apr 5(Fri), 2024

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

Abstract: In this talk, we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We will first review a direct approach to the construction of asymptotic Bergman projections, developed by Deleporte--Hitrik--Sjöstrand in the case of real analytic weights, and Hitrik--Stone in the case of smooth weights. We shall then explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, we show that Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit. We will also introduce some microlocal analysis tools in the Gevrey setting, including Borel's lemma for symbols and complex stationary phase lemma. This talk is based on joint work with Hang Xu.

Title:  Non-uniform finite-element meshes defined by ray dynamics for Helmholtz trapping problems    
Speaker: Euan Spence (University of Bath)

Speaker: Euan Spence (University of Bath)                        

Time: 8:00-9:00 pm, March 21(Thu), 2024

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


A very popular method for computing approximations to the solution of the Helmholtz equation is the finite-element method (FEM), which approximates the solution as a piecewise polynomial on a mesh. When the polynomial degree is fixed and the mesh is uniform, the question of how the meshwidth must depend on the frequency has been a classic question in numerical analysis since the 1990s, and is now well-understood. This talk introduces non-uniform meshes that allow trapping problems to be solved with fewer degrees of freedom than with a uniform mesh. These new meshes are designed using results and techniques from microlocal analysis. (No prior knowledge of the finite-element method will be assumed in the talk!) This is joint work with Martin Averseng (CNRS, Angers) and Jeff Galkowski (University College London).

Title: 调和分析及相关领域中的著名猜想(通俗报告)
Speaker: Miao Changxing 苗长兴(北京应用物理与计算数学研究所)

Time: 10:00-11:00 am, December 22(Fri), 2023

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


Besicovitch在解决Kakeya“旋针”问题过程中, 构造了Besicovitch集合( n维欧氏空间中含任意方向单位线段的集合),Fefferman率先使用Besicovitch集解决了著名的“圆盘猜想”, Bourgain天才地将Kakeya猜想提升到Kakeya极大函数猜想及Nikodym极大函数猜想,将几何测度论Kakeya猜想纳入到现代调和分析的范畴,搭建了几何测度论与调和分析研究的桥梁。该猜想的研究历经沧桑,逐步发现与Fourier限制性猜想、Bochner-Riesz猜想及局部光滑猜想密切相关,形成了调和分析领域中的著名四大猜想。 更令人惊叹的是四大猜想的研究已涉及调和分析、偏微分方程、堆垒数论、代数几何、几何测度论与关联几何、算术组合学等众多数学领域。这次报告以四大猜想为主线, 介绍与之相关的著名猜想、研究进展、研究这些猜想派生的现代数学方法,特别是介于结构性干涉与平方根消失的Bougain-Guth方法、波包分解与尺度归纳方法、decoupling理论、代数多项式分解与关联几何方法等。作为应用,还将介绍这些方法在PDEs、几何测度论、数论等研究领域的重要作用。

Title:  Topics on the essential self-adjointness for Klein-Gordon type operators on spacetimes                    
Shu Nakamura (Gakushuin University)  

Time: 10:00-11:00 am, December 8(Fri), 2023

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



We discuss recent results on the essential self-adjointnessof Klein-Gordon type operators on several classes of spacetimes. The first one is the asymptotically flat spacetime, which was studied previously by A. Vasy (J. Spectral Theory 2020) and by us (Ann. H. Lebesgue 2021), but we present a new simpler proof (Ann. H. Poincare 2023). We also discuss the essential self-adjointness for the asymptotically static spacetime, which is Cauchy compact (Comm. Math. Phys. 2023). 


Title: Inverse problem for Yang-Mills-Higgs fields
Lauri Oksanen (University of Helsinki)  

Time: 8:00-9:00 pm, November 23(Thu), 2023

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



We show that the Yang-Mills potential and Higgs field are uniquely determined (up to the natural gauge) from source-to-solution type data associated with the classical Yang-Mills-Higgs equations in the Minkowski space. We impose natural non-degeneracy conditions on the representation for the Higgs field and on the Lie algebra of the structure group which are satisfied for the case of the Standard Model. Our approach exploits non-linear interaction of waves to recover a broken non-abelian light ray transform of the Yang-Mills field and a weighted integral transform of the Higgs field. The talk is based on joint work with Xi Chen, Matti Lassas, and Gabriel Paternain. 


Title:Transport of nonlinear oscillations along rays that graze a convex obstacle to any order
Speaker: Wang Jian (University of North Carolina, Chapel Hill)  

Time: 9:00-10:00 am, November 17(Fri), 2023

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



When a wave interacts with a convex obstacle, it creates reflected waves, as well as waves in the geometric shadow of the obstacle. This phenomenon is called wave diffraction. To describe the propagation of oscillations in wave diffraction, we construct highly oscillatory asymptotic solutions to semilinear wave equations, allowing arbitrary order of grazing. As a corollary, we found that no high frequency oscillation can be observed in the geometric shadow in $H^1$ norms. This talk is based on joint work with Mark Williams.


Title: Resonances for Schr\”odinger operators on hyperbolic space
Speaker:  David Borthwick (Emory University)  

Time: 9:00-10:00 am, November 10(Fri), 2023  

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


We discuss existence and inverse results for Schr\"odinger operators on the hyperbolic space $\mathbb{H}^{n+1}$, with smooth, compactly supported potentials. Such results have been known in the Euclidean setting for several decades, based on the wave/heat trace expansions as well as asymptotics of the scatteringphase. We will present a full picture of the corresponding theory in the hyperbolic setting and highlight some of the key differences from the Euclidean case. This talk represents joint work with Yiran Wang.


Title: Geodesic Lévy Flight and the Foraging Hypothesis
Speaker: Leo Tzou (University of Armsterdam)

Time: 8:00-9:00 pm, October 26(Thu), 2023

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



The Lévy Flight Foraging Hypothesis is a widely accepted dogma which asserts that animals using search strategies allowing for long jumps, also known as Lévy flights, have an evolutionary advantage over those animals using a foraging strategy based on continuous random walks modelled by Brownian motion. However, recent discoveries suggest that this popular belief may not be true in some geometric settings. In this talk we will explore some of the recent progress in this direction which combines Riemannian geometry with stochastic analysis to create a new paradigm for diffusion process.



Title:  Nonlinear stability of the Vlasov-Poisson system in R^3

Speaker:  Wang Xuecheng (Tsinghua University)

Time:  8:00-9:00 pm, October 12(Thu), 2023

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

Abstract: We consider the stability problem for the  3D Vlasov-Poisson system in the whole space around the spatially homogeneous nontrivial equilibrium. In particular, we give linear stability for a class of general equilibrium and nonlinear stability for a special equilibrium,  which is the so-called Poisson equilibrium. This talk is based on joint works with A. Ionescu (Princeton University), B. Pausader (Brown University), and K. Widmayer (University of Zurich and University of Vienna).       


Title: Inverse Problems for Some Nonlinear PDEs with Partial Data 

Speaker:  Zhou Ting (Zhejing University)      
Time:  10:00-11:00 am, September 29(Fri), 2023      
Venue: Zoom: 618-038-6257, Password: SCMS            
In this talk, I will demonstrate the higher order linearization approach to solve several inverse boundary value problems for nonlinear PDEs, modeling for example nonlinear optics, including nonlinear magnetic Schrodinger equation and time-dependent Schrodinger equation. Considering partial data problems, the problem will be reduced to solving for the coefficient functions from their integrals against multiple linear solutions that vanish on part of the boundary. We will focus our discussion on choices of linear solutions and some microlocal analysis tools and ideas in proving injectivity of the coefficient function from its integral transforms such as the FBI transform.  
[1] R.-Y. Lai and T. Zhou, Inverse problems for nonlinear fractional magnetic Schrödinger equation, submitted, arXiv:2103.08180.  
[2] R.-Y. Lai and T. Zhou, Partial data inverse problems for nonlinear magnetic Schrödinger equation, accepted by Math. Res. Lett., arXiv:2007.02475                                                                                                  


Title: Analytic double fibration transforms

Speaker                                                                                              Mikko Salo (University of Jyväskylä)    

Time:  8:00-9:00 pm, June 9(Fri), 2023

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


We study integral transforms associated with a double fibration. This class includes various transforms encountered in tomography problems, such as (magnetic) geodesic X-ray transforms, generalized Radon transforms, and (Lorentzian) light ray transforms. If the underlying curve or surface family is real-analytic and a Bolker condition holds, we show that certain analytic singularities of a function can be determined from its transform which is treated as an analytic elliptic Fourier integral operator. This leads to local and global uniqueness results and Helgason type support theorems for these transforms. This is joint work with Marco Mazzucchelli (ENS Lyon) and Leo Tzou (Amsterdam). 


Title:                                                                     The Lorentzian scattering rigidity problem and rigidity of stationary metrics   

Speaker:  Plamen Stefanov (Purdue University)

Time:  9:00-10:00 am, May 12(Fri), 2023

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



We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation known on a lateral boundary.  We show that, under a non-conjugacy assumption,  every defining function r(x,y) of the submanifold of pairs of boundary points which can be connected by a lightlike geodesic plays the role of the boundary distance function in the Riemannian case in the following sense. Its linearization is the light ray transform of tensor fields of order two which are the perturbations of the metric. Next, we study scattering rigidity of stationary metrics in time-space cylinders and show that it can be reduced to boundary rigidity of magnetic systems on the base; a problem studied previously. This implies several scattering rigidity results for stationary metrics.


Title: Non-concentration for Neumann eigenfunctions in planar domains

Speaker: Hans Christianson (University of North Carolina, Chapel Hill)

Time: 10:00-11:00 am, Apr. 28(Fri), 2023

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



In this talk, I will discuss several new results on non-concentration of Neumann eigenfunctions in shrinking neighborhoods of planar domains.  The main result is a sharp estimate in a neighborhood of a corner point, but I will also discuss some other results in smoother situations.  The main tools are a Rellich commutator argument and an induction argument to control boundary error terms.  This is joint work with John Toth.



Title: An improved fractal Weyl upper bound in obstacle scattering

Speaker: Lucas Vacossin (Laboratoire de Mathematiques d'Orsay, Universite Paris-Saclay)

Time: 8:00-9:00 pm, Apr. 14(Fri), 2023

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



In this talk, we will be interested in the problem of scattering by strictly convex obstacles in 2D and we will give estimates for the number of resonances in regions $\{ \re \lambda \in [r, r+1], \im \lambda \geq - \gamma \}$ in the limit $r \to + \infty$. It is now well known in open quantum chaos that the dimension of the trapped set appears when counting resonances in such boxes : the number of resonances is $O(r^\delta)$ where \delta is related with the dimension of the trapped set.  This result is often called a Fractal Weyl upper bound. Nevertheless, it is expected that this bound is not optimal when $\gamma$ is below some threshold and Improved Fractal Weyl upper bounds have emerged in different contexts of open quantum chaos (convex co-compact hyperbolic surfaces, open baker's maps). In this presentation, we will explain a version of such a result in obsctacle scattering. We will explain the main ideas of the proof, which relies on a subtle interaction between the propagation of coherent states and the use of escape functions.



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.


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.



Note: The talk has been postponed.

Title: Some Theoretical and Computational Aspects of the Inverse Boundary Value Problems for the Wave Equation

Speaker:           Yang Yang (Michigan State University)

Time: TBA (updated)

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


Inverse boundary value problems (IBVPs) concern recovery of parameters in partial differential equations from boundary data. This talk will focus on the IBVPs for the wave equation. On the theoretical side, we consider the IBVP on a cylinder-like Lorentzian manifold for the Lorentzian wave equation with lower order terms. We show that local knowledge of the Dirichlet-to-Neumann map stably determines the jets of the wave parameters up to gauge transformations, and global knowledge of the map stably determines the lens relation as well as the light ray transforms of the lower order terms. On the computational side, we present a non-iterative algorithm for the acoustic IBVP to reconstruct the sound speed. The algorithm is based on the boundary control method and validated with both full and partial boundary data. The talk is based on the joint work with Plamen Stefanov and Tianyu Yang.