In the classical Calderon conjecture we want to recover a metric on a compact manifold, up to diffeomorphism fixing the boundary, from the Dirichlet-to-Neumann (DN) map of the metric Laplacian. This problem is routinely motivated by applications and is open in dimension 3 and higher. In this talk, we fix the metric and consider the DN map of the connection Laplacian for Yang-Mills connections on vector bundles. We sketch the proof of uniqueness up to gauges fixing the boundary and propose two approaches. In the first one we develop a new technique, involving degenerate unique continuation principles and an analysis of the zero set of solutions to an elliptic PDE. The second argument involves a Runge-type approximation along curves to recover holonomy and we are able to show uniqueness of both an arbitrary bundle and a Yang-Mills connection.
Combining the construction of suitable local solutions to PDEs with appropriate global approximation theorems, we can obtain global solutions to PDEs with prescribed geometric or topological properties. This technique has been used in many contexts involving elliptic equations. In this talk I will focus on showing global approximation theorems for parabolic equations and their utilisation to prove the existence of solutions of the heat equation with local hot spots with prescribed behaviour, among other applications. This is a joint work with A. Enciso and D. Peralta-Salas.
Filip Bosnic Bielefeld University, Germany
Bobby Cheng University of Sussex, United KingdomQuantum Resonances in Relativistic Systems
Significant amounts of research have been completed on mathematical quantum resonances in the non-relativistic setting. This is achieved by studying the spectra of the perturbed Schrödinger operator.However success in generalizing these results to the relativistic setting has been limited. In this poster I have outlined qualitatively some of the results of my research in this field thus far.
Dimitrije Cicmilovic University of Bonn, GermanySymplectic non-squeezing theorem in Hilbert space
Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation
Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation
We show global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. The results are based on a reduction from the fractional conductivity equation to the fractional Schrödingerequation.
A NON LOCAL PROBLEM OF DIFFERENTIAL EQUATIONS VIA NON LOCAL OPERATORS OF ARBITRARY ORDER
In the proposed poster, we are concerned with a nonlocal problemfor a class of differential equations using nonlocal fractional operators in Banachspaces. Some existence results of positive solutions are obtained. Then, thestability of the problem is discussed. Finally, to illustrate the effectiveness ofthe results obtained, some example are presented
New mixed operators of Hadamard type with applications
In the 2d poster, we establish recent results on the Hadamard operators. Some mixed integral operators, that generalize those of Hadamard type, are delivered and some of their properties are proved. Finally,some applications are discussed.
Guy Fabrice Foghem Gounoue Bielefeld University, GermanyInhomogeneous Neumann Problem for the fractional Laplace operator
Robin Görmer Universität Bremen, Germany3D Electrical Impedancetomography
Andreea Grecu University of Bucharest, RomaniaThe Schrodinger equation on star-graphs under general coupling conditions
Yulia Meshkova St. Petersburg State University, RussiaOperator error estimates for homogenization of elliptic and parabolic systems
We consider a matrix strongly elliptic second order differential operator acting in a bounded domain with the Dirichlet boundary condition. The operator is self-adjoint. Coefficients are periodic and oscillate rapidly. We study the behavior of solutions of the corresponding elliptic and parabolic systems in the small period limit. The results can be written as approximations of the resolvent and the semigroup in L2→L2 and L2→H1 operator norms. So, the estimates of this type are called operator error estimates in homogenization theory. The talk is based on a joint work with T. A. Suslina.
Mihai Nechita University College London, United KingdomUnique continuation for the Helmholtz equation. Stability estimates and numerical analysis
We consider the unique continuation problem for the Helmholtz equation that arises, e.g. in inverse boundary value problems for the wave equation. Following previous work by Isakov we prove conditional Hölder stability estimates with constants independent of the wave number, when the solution is continued along a convex surface. The main tools we employ are Carleman estimates and semiclassical analysis.We then introduce a stabilized finite element method and prove convergence with the order given by the conditional stability, and with explicit dependence on the wave number.
Andrea Nickel Bielefeld University, DeutschlandA nonlocal maximum principle
Leyter Potenciano Machado University of Jyväskylä, FinlandA resolvent estimate for the magnetic Schrödinger operator in the presence of short and long-range
Jesse Railo University of Helsinki & University of Jyväskylä, FinlandTensor tomography on Cartan–Hadamard manifolds
We study the geodesic x-ray transform on Cartan–Hadamard manifolds, generalizing the x-ray transforms on Euclidean and hyperbolic spaces that arise in medical and seismic imaging. We prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity.
Yafet Sanchez Sanchez Max Planck Institute-Bonn, GermanyPropagators on spacetimes of low regularity
Vanja Wagner Universität Bielefeld, GermanyBoundary behaviour of certain pure-jump Markov processes on sets
Filip Bosnic Bielefeld University, Germany
Bobby Cheng University of Sussex, United KingdomQuantum Resonances in Relativistic Systems
Significant amounts of research have been completed on mathematical quantum resonances in the non-relativistic setting. This is achieved by studying the spectra of the perturbed Schrödinger operator.However success in generalizing these results to the relativistic setting has been limited. In this poster I have outlined qualitatively some of the results of my research in this field thus far.
Dimitrije Cicmilovic University of Bonn, GermanySymplectic non-squeezing theorem in Hilbert space
Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation
Giovanni Covi Jyväskylän Yliopisto, FinlandInverse problems for a fractional conductivity equation
We show global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. The results are based on a reduction from the fractional conductivity equation to the fractional Schrödingerequation.
A NON LOCAL PROBLEM OF DIFFERENTIAL EQUATIONS VIA NON LOCAL OPERATORS OF ARBITRARY ORDER
In the proposed poster, we are concerned with a nonlocal problemfor a class of differential equations using nonlocal fractional operators in Banachspaces. Some existence results of positive solutions are obtained. Then, thestability of the problem is discussed. Finally, to illustrate the effectiveness ofthe results obtained, some example are presented
New mixed operators of Hadamard type with applications
In the 2d poster, we establish recent results on the Hadamard operators. Some mixed integral operators, that generalize those of Hadamard type, are delivered and some of their properties are proved. Finally,some applications are discussed.
Guy Fabrice Foghem Gounoue Bielefeld University, GermanyInhomogeneous Neumann Problem for the fractional Laplace operator
Robin Görmer Universität Bremen, Germany3D Electrical Impedancetomography
Andreea Grecu University of Bucharest, RomaniaThe Schrodinger equation on star-graphs under general coupling conditions
Yulia Meshkova St. Petersburg State University, RussiaOperator error estimates for homogenization of elliptic and parabolic systems
We consider a matrix strongly elliptic second order differential operator acting in a bounded domain with the Dirichlet boundary condition. The operator is self-adjoint. Coefficients are periodic and oscillate rapidly. We study the behavior of solutions of the corresponding elliptic and parabolic systems in the small period limit. The results can be written as approximations of the resolvent and the semigroup in L2→L2 and L2→H1 operator norms. So, the estimates of this type are called operator error estimates in homogenization theory. The talk is based on a joint work with T. A. Suslina.
Mihai Nechita University College London, United KingdomUnique continuation for the Helmholtz equation. Stability estimates and numerical analysis
We consider the unique continuation problem for the Helmholtz equation that arises, e.g. in inverse boundary value problems for the wave equation. Following previous work by Isakov we prove conditional Hölder stability estimates with constants independent of the wave number, when the solution is continued along a convex surface. The main tools we employ are Carleman estimates and semiclassical analysis.We then introduce a stabilized finite element method and prove convergence with the order given by the conditional stability, and with explicit dependence on the wave number.
Andrea Nickel Bielefeld University, DeutschlandA nonlocal maximum principle
Leyter Potenciano Machado University of Jyväskylä, FinlandA resolvent estimate for the magnetic Schrödinger operator in the presence of short and long-range
Jesse Railo University of Helsinki & University of Jyväskylä, FinlandTensor tomography on Cartan–Hadamard manifolds
We study the geodesic x-ray transform on Cartan–Hadamard manifolds, generalizing the x-ray transforms on Euclidean and hyperbolic spaces that arise in medical and seismic imaging. We prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity.
Yafet Sanchez Sanchez Max Planck Institute-Bonn, GermanyPropagators on spacetimes of low regularity
Vanja Wagner Universität Bielefeld, GermanyBoundary behaviour of certain pure-jump Markov processes on sets
I will talk about the inverse Calderón problem, which consists of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually assumes the data to be given by such a map. This situation corresponds to having access to infinite-precision measurements, which is totally unrealistic. In this lecture, we will discuss the Calderón problem assuming the data to contain measurement errors and provide formulas to reconstruct the conductivity and its normal derivative on the surface.
In this course we will present basic tools for the study of high-frequency eigenfunctions or quasimodes of Schrödinger-type elliptic operators on a compact manifold. Our mail goal will be to give a quantitative description of concentration and oscillation effects that can be developed by a sequence of quasimodes, and in particular to show how this issue is related the global geometry of the manifold. We will give applications to unique continuation-type estimates and to the Calderón problem. We will mainly use tools of from the theory of pseudodifferential operators and semiclassical measures that will be reviewed during the course.
In this course we will present basic tools for the study of high-frequency eigenfunctions or quasimodes of Schrödinger-type elliptic operators on a compact manifold. Our mail goal will be to give a quantitative description of concentration and oscillation effects that can be developed by a sequence of quasimodes, and in particular to show how this issue is related the global geometry of the manifold. We will give applications to unique continuation-type estimates and to the Calderón problem. We will mainly use tools of from the theory of pseudodifferential operators and semiclassical measures that will be reviewed during the course.
The goal of this talk is to study bounds on the Riesz means of mixed Steklov problems. The Riesz mean is a convex function of eigenvalues and has an important role and connection with other spectral quantities. We recall the results known in this direction for the Laplace eigenvalues. Then we introduce the mixed Steklov problem and state the main results. We also discuss some key ideas of the proof. This is joint work with Ari Laptev.
This talk is concerned with the inverse spectral problem of recovering the magnetic field and the electric potential in a Riemannian manifold from some asymptotic knowledge of the boundary spectral data of the corresponding Schrödinger operator with Dirichlet (or Neumann) boundary conditions. It combines a representation formula coming from Isozaki as well as a construction of solutions to the Schrödinger equation in simple manifolds already used by Bellassoued and myself in the quantitative study of inverse problems on the Dirichlet-to-Neumann map associated with evolution equations (also inspired by the study of the anisotropic Calderón problem by Kenig, Salo, Uhlmann and myself).
This is a joint work with Mourad Bellassoued, Mourad Choulli, Yavar Kian and Plamen Stefanov.
In this course we will present basic tools for the study of high-frequency eigenfunctions or quasimodes of Schrödinger-type elliptic operators on a compact manifold. Our mail goal will be to give a quantitative description of concentration and oscillation effects that can be developed by a sequence of quasimodes, and in particular to show how this issue is related the global geometry of the manifold. We will give applications to unique continuation-type estimates and to the Calderón problem. We will mainly use tools of from the theory of pseudodifferential operators and semiclassical measures that will be reviewed during the course.