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Principal Speakers

Gregory Berkolaiko (Texas A&M)
Interlacing eigenvalue inequalities and counting zeros of graph eigenfunctions

Dorin Bucur (Chambéry)
Optimization and spectral inequalities

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Abstract: In these lectures, isoperimetric inequalities involving the spectrum of the Laplace operator will be seen from a shape optimisation point of view. I will present recent results based on free boundary techniques which can be used to obtain qualitative information on the domains minimising a spectral functional. As main example, we shall focus on the minimisation of the k-th eigenvalue of the Laplace operator with Dirichlet boundary conditions, under a volume constraint. I will also show how existence and regularity of an optimal shape for a (very) particular class of functionals, implies that the shape is a ball! For example, this argument works for the first eigenvalue of the Dirichlet Laplacian, and gives a proof of the Faber-Krahn inequality which does not use rearrangements.

Bruno Colbois (Neuchâtel)
The spectrum of the Laplacian: a geometric approach

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Abstract: The goal of these lectures is to present different aspects of the spectrum of the Laplacian on a compact Riemannian manifold from a geometric viewpoint. In the first part, I recall without proofs some classical facts, try to illustrate the theory by examples and in discussing open questions. In the second part, I focus on a specific problem: the estimate of the spectrum in the conformal class of a given Riemannian metric. This is the opportunity to explain some interesting geometrical methods and to present recent developments on the subject.

Chen Greif (UBC)

Numerical Solution of Linear Eigenvalue Problems

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Abstract: We review numerical methods for computing eigenvalues and singular values of matrices. We start by considering the computation of the dominant eigenpair of a general dense matrix using the power method, and then generalize to orthogonal iterations and the QR iteration with shifts. We also consider divide-and-conquer algorithms for tridiagonal matrices. The second part of the course involves the computation of eigenvalues of large and sparse matrices. The Lanczos and Arnoldi methods are developed and described within the context of Krylov subspace eigensolvers. The Golub-Kahan bidiagonalization is also described, for computing or approximating singular values. The algorithms are illustrated by numerical experiments, using Matlab.

Daniel Grieser (Oldenburg)
Asymptotics of eigenvalues on thin things

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Abstract: I will explain methods to obtain complete asymptotic expansions of eigenvalues of the Laplacian on domains which are close to one-dimensional, for example fat graphs and thin triangles.

Colin Guillarmou (ENS)
A scattering theory approach for X-ray tomography

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Abstract: Using ideas coming from scattering theory, we discuss the problem of injectivity of the X-ray transform and application to boundary rigidity problems. The X-ray transform is the transform which associates to a function the set of its integrals along all possible geodesics relating boundary points on a Riemannian manifold with boundary. Properties of this transform are related to solvability of certain transports equations. A surprising feature originally due to Pestov-Uhlmann is that in dimension 2, there are strong connections between the Dirichlet-to-Neumann operator for the laplacian and X-ray transform. In this course we shall discuss what happens also when there exist trapped geodesics in the manifold.

Bernard Helffer (Paris-Sud)
On nodal partitions and minimal spectral partitions (an introduction)

Guido Kanschat (Heidelberg)

Finite element approximation of eigenvalue problems

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Abstract: We will begin with a review of basic principles of finite element theory. Then, we study the finite element approximation of self-adjoint positive definite eigenvalue problems and Rayleigh quotients. We discuss constrained eigenvalue problems and the phenomenon of spurious eigenvalues. Finally, we will consider compatible discretizations of vector Laplacian and Maxwell eigenvalue problems. In practical sessions, we run numerical experiments with the finite element software deal.II.

Richard Melrose (MIT)
Laplacians degenerating at a point and gluing

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Abstract: I will talk about the behaviour of the Laplace operator for metrics degenerating at a point, especially as this is related to "gluing constructions" in Riemannian geometry.

Richard Schoen (Stanford)
The spectral geometry of the Dirichlet-Neumann operator

Mikhail Sodin (Tel Aviv)
Random Nodal Portraits

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Abstract: We describe the progress and challenges of understanding the zero sets of smooth Gaussian random functions of several real variables. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution. This might be thought as a statistical version of Hilbert's 16th problem. The lectures will be based on joint works with Fedor Nazarov.

Alexander Strohmaier (Loughborough)

Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

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Part 1- Method of Particular Solutions. I will explain the method of particular solutions and how it can be used to compute eigenvalues of the Laplace operator on Riemannian manifolds. I will demonstrate this on hyperbolic surfaces and show how eigenvalues can be computed with very high accuracy in this case.

Part 2 - Computations of Spectral Invariants. I will describe how a combination of analytic and numerical tools can be used to compute spectral invariants such as the spectral Zeta function and the Zeta regularized determinant of the Laplace operator. I will also demonstrate some ways how regularized Weyl laws can be useful for numerical computations.

Part 3 - Dirichlet-to-Neumann map. I will discuss the Dirichlet-to-Neumann map for Riemannian manifolds with boundary and how it can be used to find resonances for non-compact manifolds.