In alphabetical order by last name. Please refer to the programme for the ordering of talks.
Synopsis: In two dimensions, many geometric inverse problems have a complex geometric flavour. Transport twistor spaces provide a means for reasoning about this in a precise way. In the mini-course we will explore this idea, discuss where thinking about twistor space can be useful, and where new challenges arise.
Synopsis: In this minicourse, we will provide an introduction to the field of inverse problems for elliptic PDEs, covering both classical results and recent advancements. We will start with the renowned Calderon problem, which seeks to determine the electrical conductivity of a medium from voltage and current measurements on its boundary. This problem forms the basis for Electrical Impedance Tomography, an imaging modality with applications in seismic and medical imaging. We will present global uniqueness results for this and related problems, addressing both full and partial data cases, using Carleman estimate techniques.
Next, we will explore inverse boundary problems for elliptic PDEs in transversally anisotropic Riemannian geometries. We will also address inverse problems for nonlinear elliptic PDEs, both in Euclidean domains and on Riemannian manifolds. In particular, we will see that the presence of nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart remains open. Finally, we will discuss inverse problems for nonlocal elliptic operators on compact Riemannian manifolds without boundary. Throughout the minicourse, we will also state several open problems in the field.
Marco Mazzucchelli (École Normale Supérieure - Lyon): "Boundary and length spectrum rigidity problems in Riemannian manifolds"
Synopsis: In this series of three lectures, I will introduce a few, related, geometric inverse problems involving geodesics on Riemannian manifolds: on a compact Riemannian manifold with boundary, to what extend does the Riemannian distance function among boundary points determine the inner geometry? On a compact Riemannian manifold without boundary, to what extend does the set of lengths of the closed geodesics, possibly marked with extra data, determine the geometry? In the course of the lectures, I will make precise formulations of these problems, and present a selection of celebrated results, from classical all the way to very recent ones, and list a few open problems.
Tentative program:
1) Introduction to the boundary rigidity and marked length-spectrum rigidity problems
2) Rigidity results in negative curvature
3) Rigidity results for Anosov closed Riemannian surfaces and open problems
The lectures are based on the upcoming book:
https://perso.ens-lyon.fr/marco.mazzucchelli/preprints/inverse_problems.html
Abstract: In this talk I will consider the inverse spectral problem of determining a Riemannian metric and a magnetic field from the Dirichlet-to-Neumann (DN) map of a magnetic Laplacian on a compact surface (magnetic Steklov problem). We will first present a sharp spectral asymptotics result for the eigenvalues of the DN map (up to arbitrary polynomial precision) and hence derive spectral invariants. Up to lower order terms, the spectrum looks like a union of arithmetic progressions, one for each boundary component. We will then discuss if this information determines the number of boundary components, their lengths, parallel transport and magnetic flux on each of them (encoded in the coefficients of the arithmetic progressions), and relate this to covering systems arising in number theory. We will give examples of distinct surfaces and magnetic potentials which have the same spectrum up to small error, and present stronger rigidity results in the case of zero magnetic field. Joint work with Anna Siffert.
Alexis Drouot (University of Washington, Seattle): "Topological insulators in semiclassical regime."
Abstract: We will study the dynamics of coherent states near interfaces between topological insulators in the semiclassical regime. We will show that these states split in two parts: one that propagates coherently at a predetermined speed and direction, and one that immediately collapses. This collapse is really unusual in the study of coherent states. Our approach relies on semiclassical normal forms for 2x2 symbols with crossings of eigenvalues.
Abstract: Nonlinearity in Kinetic models often characterizes important physical properties; meanwhile, it also introduces difficulty in solving both direct and inverse problems. In this talk, we will introduce mathematical background and several interesting applications for inverse problems. Moreover, we will address the linearization technique that recently has been successfully employed to recover the coefficients uniquely and stably in a wide variety of nonlinear PDEs. The focus will be on applying this technique to determine nonlinear terms in the transport equations.
Fatma Terzioglu (North Carolina State University): "Analytic inversion of an integral transform arising in Compton camera imaging"
Abstract: Recent advancements in imaging have led to several new types of Radon-type transforms, involving integration over a family of conical surfaces termed cone (or Compton) transforms (V-line or broken ray transforms in 2D). These transforms arise prominently in Compton camera imaging, which finds applications in astronomy, medical diagnostics, and homeland security. In this talk, we introduce Compton camera imaging and address the analytic inversion of the cone transform via relations with the Radon transform.
Nikolas Eptaminitakis (Leibniz University Hannover): "The Hyperbolic X-Ray Transform: Range Characterizations, Functional Relations and Mapping Properties"
Abstract: In this talk we will discuss some new results regarding the geodesic X-ray transform on 2-dimensional hyperbolic space, mainly focusing on range characterizations, functional relations with differential operators of wedge type, and mapping properties between suitable scales of Sobolev spaces. Our approach makes use of recent analogous results for the X-ray transform on the Euclidean disk, together with the projective equivalence between the latter and the Beltrami-Klein model of hyperbolic space. Based on joint work with François Monard and Yuzhou (Joey) Zou.
Boya Liu (North Dakota State University): "Recovery of time-dependent coefficients in hyperbolic equations on Riemannian manifolds from partial data"
Abstract: In this talk we discuss inverse problems of determining time-dependent coefficients appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines time-dependent coefficients of the wave equation uniquely in a space-time cylinder. We shall discuss two problems: (1) Recovery of a potential appearing in the wave equation, when the Dirichlet and Neumann values are measured on opposite parts of the lateral boundary of the space-time cylinder. (2) Recovery of both a damping coefficient and a potential appearing in the wave equation, when the Dirichlet values are measured on the whole lateral boundary and the Neumann data is collected on roughly half of the boundary. This talk is based on joint works with Teemu Saksala (NC State University) and Lili Yan (University of Minnesota).
Abstract: In this talk we’ll discuss exponential localization of Schrodinger eigenfunctions in the presence of both scalar and vector potentials (also known as electric and magnetic potentials). After recalling Agmon’s celebrated method of obtaining exponential decay estimates away from the ’classically forbidden region’, we discuss difficulties in this approach in the magnetic case, as well as recent work in the 2d case.
Abstract: For planar billiard tables, the marked length spectrum encodes the lengths of action minimizing orbits having a given rational rotation number. For strictly convex tables, a renormalization of these lengths extends to a continuous function called Mather’s beta function (or the mean minimal action). We show that using the algebraic structure of its Taylor coefficients, one can prove C∞ compactness of marked length isospectral sets. This gives a dynamical analogue of the Laplace spectral results of Melrose, Osgood, Phillips and Sarnak.
William Trad (Hong-Kong University of Science and Technology): "Mean sojourn time and eigenvalue asymptotic expansions"
Abstract: Within this talk, we will explore how the theory of pseudo-differential operators is used to construct explicit Greens function expansions in order to derive mean sojourn time asymptotic expansions for a Brownian motion confined within a domain. In addition, we seek to describe how these Greens function expansions can be used in deriving a spectral asymptotic result on the variation of eigenvalues. What appears ubiquitously throughout our results are geometric quantities such as the mean and principal curvatures, as well as the geodesic distance and volumes of the geometries in question.
Lili Yan (University of Minnesota): "Stable determination of time-dependent collision kernel in the nonlinear Boltzmann equation."
Abstract: In this talk, we consider an inverse problem for the nonlinear Boltzmann equation with a time-dependent kernel in dimensions n ≥ 2. We establish a logarithm-type stability result for the collision kernel from measurements under certain additional conditions. A uniqueness result is derived as an immediate consequence of the stability result. Our approach relies on second-order linearization, multivariate finite differences, as well as the stability of the light-ray transform. This is based on joint work with Ru-Yu Lai.
Xiaowen Zhu (University of Washington, Seattle): "Bulk-edge correspondence for topological insulator with curved interface"
Abstract: Topological insulators are central objects in condensed matter physics. They refer to insulating phases of matter (i.e. the evolution is described by a Hamiltonian with a spectral gap) to which one can associate a non-trivial topological invariant. When two insulators with distinct topological invariants are glued together, protected gapless currents emerge along the interface: the material becomes a conductor along its edge. Furthermore, the edge conductance is quantized and it equals the difference of the bulk topological invariants for straight interfaces. This fundamental result is called the bulk-edge correspondence. In this talk, we discuss bulk-edge correspondence for topological insulators with curved interface. This is based on a joint work with Alexis Drouot.
Abstract: I will discuss the artificial boundary/convex foliation method, used by de Hoop, Stefanov, Uhlmann, Vasy, et al. to solve problems in X-ray tomography, boundary rigidity, and elastic travel time tomography. The analysis behind this method reduces to finding an appropriate pseudodifferential calculus, where ”elliptic” operators (such as an appropriately modified version of a normal operator for the X-ray transform) have a nice inversion theory. I will also discuss the case of recovering material parameters from travel time data in transversely isotropic elasticity, where the pertinent operators have degenerate ellipticity and more closely resemble parabolic operators; the analysis involved is more subtle and requires a more careful construction of an appropriate pseudodifferential calculus.