Schedule for: 19w5238 - Probing the Earth and the Universe with Microlocal Analysis

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We will remember the life and works of Slava Kurylev who recently passed away.

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.

We study the isotropic elastic wave equation in a bounded domain with boundary with coefficients having jumps at a nested set of interfaces satisfying the natural transmission conditions there. We analyze in detail the microlocal behavior of such solution like reflection, transmission and mode conversion of S and P waves, evanescent modes, Rayleigh and Stoneley waves. In particular, we recover Knott's equations in this setting. We show that knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the P and the S waves if there is a strictly convex foliation with respect to them, under an additional condition of lack of full internal reflection of some of the waves. This is a joint work with Andras Vasy and Gunther Uhlmann

I will describe an extension to the boundary of the cosphere bundle and geodesic flow of an asymptotically hyperbolic manifold. I will then discuss injectivity results for X-ray transforms on tensors and the boundary rigidity problem of determining an asymptotically hyperbolic metric from the renormalized lengths of geodesics joining boundary points. One part is work with Colin Guillarmou, Plamen Stefanov and Gunther Uhlmann and another part is work with Nikolas Eptaminitakis.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Analytically computing the spectrum of the Laplacian is impossible for all but a handful of classical examples. Consequently, it can be tricky business to determine which geometric features are spectrally determined; such features are known as geometric spectral invariants. Weyl demonstrated in 1912 that the area of a planar domain is a geometric spectral invariant. In the 1950s, Pleijel proved that the n-1 dimensional volume of a smoothly bounded n-dimensional Riemannian manifold is a geometric spectral invariant. Kac, and McKean & Singer independently proved in the 1960s that the Euler characteristic is a geometric spectral invariant for smoothly bounded domains and surfaces. At the same time, Kac popularized the isospectral problem for planar domains in his article, ``Can one hear the shape of a drum?'' Colloquially, one says that one can ``hear'' spectral invariants. In this talk I will not only discuss my work with many collaborators (Rafe Mazzeo, Zhiqin Lu, Clara Aldana, Klaus Kirsten, David Sher, Medet Nursultanov) but also highlight the works of many other colleagues, who share a similar interest in ``hearing singularities.''

The concentration properties of eigenfunctions of the Laplace-Beltrami operator are closely linked the the underlying geometry (and dynamics) on manifolds. It this talk I will discuss the known concentration results and the models we use to test sharpness. Such problems are effectively forward problems, I will also discuss what an inverse problem would look like in this setting.

A closed Riemannian manifold is said to be Anosov if its geodesic flow on its unit tangent bundle is Anosov (also called uniformly hyperbolic in the literature). Typical examples are provided by negatively-curved manifolds. On such manifolds, the X-ray transform is simply defined as the integration of continuous functions along periodic geodesics. I will review some recent results on the analytic study of the X-ray transform (in particular, stability estimates). The techniques rely on microlocal tools introduced by Guillarmou and further investigated by Guillarmou-Lefeuvre, and on new finite and approximate Livsic theorems proved by Gouëzel-Lefeuvre. If time permits, I will explain how these results can be applied to prove the local rigidity of the marked length spectrum.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe a new energy estimate for the geodesic vector field of a manifold of negative curvature. The estimate has several applications including injectivity of non-abelian X-ray transforms. This is joint work with Mikko Salo.

We consider the Dirichlet-to-Neumann map, defined in a suitable sense, for the equation $-\Delta u + V(x,u)=0$ on a compact Riemannian manifold with boundary. We show that, under certain geometrical assumptions, the Dirichlet-to-Neumann map determines V for a large class of non-linearities. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex geometric optics solutions for the linearized operator, and the resulting non-linear interactions. This approach allows us to reduce the inverse problem boundary value problem to the purely geometric problem to invert a family of weighted ray transforms, that we call the Jacobi weighted ray transform. This is a joint work with Ali Feizmohammadi.

Given an $L^2$-normalized eigenfunction with eigenvalue $\lambda^2$ on a Riemannian manifold $(M,g)$ and a nonempty open set $\Omega\subset M$, what lower bound can we prove on the $L^2$-mass of the eigenfunction on $\Omega$? The unique continuation principle gives a bound for any $\Omega$ which is exponentially small as $\lambda\to\infty$. On the other hand, microlocal analysis gives a $\lambda$-independent lower bound if $\Omega$ is large enough, i.e. it satisfies the geometric control condition. This talk presents a $\lambda$-independent lower bound for any set $\Omega$ in the case when $M$ is a hyperbolic surface. The proof uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from harmonic analysis called the Fractal Uncertainty Principle. Applications include control for Schrödinger equation and exponential decay of damped waves. Joint work with Jean Bourgain, Long Jin, and Joshua Zahl.

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, based on a generalization of the transmission condition, allows to prove a Bernstein–von Mises theorem, about the statistical reliability of the Maximum A Posteriori 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 a consistency result for the mean of the posterior distribution in the large data sample limit. Numerical illustrations will be presented. Joint works with Gabriel Paternain and Richard Nickl (Cambridge).

The singularities of solutions of the elastic wave equation follow a certain flow on cotangent bundle. For a typical anisotropic stiffness tensor this not the cogeodesic of a Riemannian geometry. But with a tiny additional assumption the singularities of the fastest polarization do correspond to a Finsler geometry. I will discuss the arising geometrical structure and some recent results in Finsler geometry arising from elasticity.

In this talk, we consider the following question: Given the source to solution map for a relativistic Boltzmann equation on a known open set $V$ of a Lorentzian spacetime $(\mathbb{R}\times N,g)$, can we this data to uniquely determine the spacetime metric on an unknown region of $\mathbb{R}\times N$? We will show that the answer is yes. Precisely, we determine the metric up to conformal factor on the domain of causal influence for the set $V$. Key to our proof is that the nonlinearity in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linearized problem. We use this relationship together with an analysis of the behaviour of particle collisions by microlocal techniques to determine the set of locations in $V$ where we first receive light signals from collisions in the unknown domain. From this data we obtain the desired diffeomorphism. The strategy of using the nonlinearity of the inverse problem as a feature with which to gain knowledge of a related linearized problem is classical (see for example [2]). In a Lorentzian setting, this technique combined with microlocal analysis first appeared in [1] in the context of a wave equation with a quadratic nonlinearity and source-to-solution data. We will briefly survey this and later related work as they provide context for our result. We will also provide some physical motivation and context for the problem we consider. The new results presented in this talk is joint work with Antti Kujanapää, Matti Lassas, and Tony Liimatainen (University of Helsinki). [1] Kuylev Y., Lassas M., Uhlmann G., Inventiones mathematicae 212.3 (2018): 781-857. [2] Sun Z., Mathematische Zeltschrift 221 (1996): 293-305.

We consider a bounded domain $\Omega \subset \mathbb{R}^3$ on which Lam\'{e} parameters are piecewise smooth. We consider the elastic wave initial value inverse problem, where we are given the solution operator for the elastic wave equation, but only outside $\Omega$ and only for initial data supported outside $\Omega$. Using our recently introduced scattering control series in the acoustic case, and a layer stripping argument, we prove that piecewise smooth Lam\'{e} parameters are uniquely determined by this map. We make use of microlocal analysis to avoid using unique continuation results but require a convex foliation condition, introduced by Uhlmann and Vasy, for both the \textit{P}- and \textit{S}-wave speeds. Joint research with P. Caday, G. Uhlmann and V. Katsnelson.

We address a geometric inverse problem: Consider a simply connected Riemannian 3-manifold (M,g) with boundary. Assume that given any closed loop \gamma on the boundary, one knows the area of the area-minimizer bounded by \gamma. Can one reconstruct the metric g from this information? We answer this in the affirmative in a very broad open class of manifolds, notably those that admit sweep-outs by minimal surfaces from all directions. We will briefly discuss the relation of this problem with the question of reconstructing a metric from lengths of geodesics, and also with the Calderon problem of reconstructing a metric from the Dirichlet-to-Neumann operator for the corresponding Laplace-Beltrami operator. Connections with this question in the AdS-CFT correspondence will also be made. Joint with T. Balehowsky and A. Nachman.

We discuss the partial data inverse boundary problem for the Schrodinger operator at a fixed frequency on a bounded domain in the Euclidean space, with impedance boundary conditions. Assuming that the potential is known in a neighborhood of the boundary, the knowledge of the partial Robin-to-Dirichlet map along an arbitrarily small portion of the boundary determines the potential uniquely, in a logarithmically stable way. In this talk we show that the logarithmic stability can be improved to the one of Holder type in the high frequency regime. Our arguments are based on boundary Carleman estimates for semiclassical Schrodinger operators acting on functions satisfying impedance boundary conditions. This is joint work with Gunther Uhlmann.

Semiclassical resolvent estimates are important for their applications to scattering theory and wave decay. The norm of the resolvent depends on dynamical properties of the bicharacteristic flow at the trapped set (the set of bicharacteristics that remain in a compact set for all time). When trapping is mild in an appropriate sense, the resolvent norm is only large at those points of phase space where trapping actually occurs. But when trapping is not mild, simple examples show that the resolvent norm may be large arbitarily far away from the trapped set. In this talk I will focus mostly on the one-dimensional and radial cases, where some optimal results are known. I will also discuss some related results in more general geometric situations. This talk is based on joint work with Long Jin and with Jacob Shapiro.

We study the local propagation of singularities of solutions of $P(y,D) u= f(y,u),$ in $R^3,$ where $P(y,D)$ is a second order strictly hyperbolic operator and $f\in C^\infty.$ We choose a time function $t$ for $P(y,D)$ and assume that $f(y,u)$ is supported on $t>-1$ and that for $t<-2,$ $u$ is assumed to be the superposition of three conormal waves that intersect transversally at a point $q$ with $t(q)=0.$ We show that, provided the incoming waves are elliptic conormal distributions of appropriate type and $(\p_u^3 f)(q, u(q))\not=0,$ the nonlinear interaction will produce singularities on the light cone for $P$ over $q.$ Melrose and Ritter, and Bony, had independently shown that the solution $u$ is a Lagrangian distribution of an appropriate class associated with the light cone over $q$ and we show that under this non-degeneracy condition, $u$ is an elliptic Lagrangian distribution and we compute its principal part.

Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space. This talk is based on joint work with Jeremy Marzuola, Andras Vasy, and Jared Wunsch.

We consider the motion of a classical colored spinless particle under the influence of an external Yang-Mills potential $A$ on a compact manifold with boundary of dimension $\geq 3$. We show that under suitable convexity assumptions, we can recover the potential $A$, up to gauge transformations, from the lens data of the system, namely, scattering data plus travel times between boundary points. This is joint work with Gabriel Paternain and Gunther Uhlmann.

I will discuss work I have done applying microlocal analysis to Compton scattering tomography and the geodesic ray transform. These are different topics linked by the common use of microlocal analysis. In Compton scattering tomography I will discuss the analysis of the normal operator in a particular scanning geometry which shows that the normal operator in this case is the sum of paired Lagrangian operators. The Lagrangian which is not the diagonal is explicitly found, and the results are supported with numerical demonstrations. For geodesic ray transform, I will discuss how to characterise of the strength of artifacts occurring at conjugate points, in two dimensions, in terms of vanishing Jacobi fields.

We consider the inverse problem of determining uniquely an expression appearing in a, linear or non-linear, diffusion equation. In the linear case, our equation is a convection-diffusion type of equation describing the transfer of mass, energy and other physical quantities. Our inverse problem consists in determining the velocity field associated with the moving quantities as well as information about the density of the medium. We consider this problem in a general setting where we associate the information under consideration with non-smooth coefficients depending on time and space variables. In the non-linear case, we treat the determination of a quasi-linear term appearing in a non-linear diffusion equation. This talk is based on a joint work with Pedro Caro.

In this talk I present global existence results backwards from scattering data for various semilinear wave equations on Minkowski space satisfying the (weak) null condition.​ These models are motivated by the Einstein​ equations in harmonic gauge, and the data is given in the form of the radiation field. It is shown in particular that the solution has the same spatial decay as the radiation field along null infinity. I will discuss the proof which relies, on one hand on a​ fractional Morawetz estimate, and on the other hand on the construction of suitable approximate​ solutions from the scattering data.

Earthquakes produce seismic waves. They provide a way to obtain information about the deep structures of our planet. The typical measurement is to record the travel time difference of the seismic waves produced by an earthquake. If the network of seismometers is dense enough and they measure a large number of earthquakes, we can hope to recover the wave speed of the seismic wave from the travel time differences. In this talk we will consider geometric inverse problems related to different data sets produced by seismic waves. We will state some uniqueness results for these problems and consider the mathematical tools needed for the proofs. The talk is based on joint works with: Maarten de Hoop, Joonas Ilmavirta, Matti Lassas and Hanming Zhou.

This talk will have a dynamical and an analytical component. On the dynamical side, we will study the properties of the billiard flow on $3$ dimensional convex polyhedra. More precisely, we will study periodic broken geodesics not hitting a neighbourhood of the $1$-skeleton of the boundary (also called ``pockets"). On the analytical side, we will apply these results to prove a quantitative Laplace eigenfunction mass concentration near the pockets, using semiclassical tools and control-theoretic results. This is joint work with B. Georgiev and M. Mukherjee.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.