Schedule for Lisbon

Monday, 8 June 2015

8:50 Announcements and information
9-9:50 Dorothee Schueth: Inaudibility of sixth order curvature invariants
10-10:50 Victor Guillemin: Semi-classical weights and equivariant spectral theory
11:20-12:10 Zuoqin Wang: Hearing symmetric potentials for toric varieties
***lunch break***
13:30-14:20 Stuart Hall: Direct spectral problems for toric Kähler-Einstein metrics
Note that there are some seminars in the general area of quantization that will take place at Técnico on Monday after our talks end. We will announce the details of those seminars in the "Announcements" session on Monday morning.

Tuesday, 9 June 2015

9-9:50 Iosif Polterovich: Billiards with a large Weyl remainder
10-10:50 David Sher: Nodal length of Steklov eigenfunctions
11:20-12:10 Pedro Freitas: The first eigenvalue of the Laplacian with Robin boundary conditions
***lunch break***
Travel to Porto

Schedule for Porto

Wednesday, 10 June 2015

9:30-10:20 Juan Gil: Spectra of singular elliptic operators on graphs
***coffee break***
11-11:50 Eveline Legendre: The first eigenvalue of compact toric Kähler manifold

Thursday, 11 June 2015

9:30-10:20 Carolyn Gordon: Isospectrality of Dirichlet-to-Neumann operators
***coffee break***
11-11:50 Alexandre Girouard: The Steklov spectrum: asymptotics and invariants
***lunch break***
13:30-14:20 Craig Sutton: Hearing the length spectrum of a compact Lie group
14:30-15:20 Alejandro Uribe: The asymptotic pseudospectrum and numerical range of complex potentials on Zoll manifolds

Abstracts

Pedro Freitas: We present recent results regarding Bareket's 1977 conjecture which states that the ball maximises the first eigenvalue of the Laplacian with Robin boundary conditions. This is joint work with David Krecirik and Pedro Antunes.

Juan Gil: In this talk I will discuss some spectral properties of second order regular singular operators on a quantum graph. More specifically, I will focus on the resolvent decay of elliptic operators with a singular potential of Coulomb type. Our approach follows some techniques developed for the study of elliptic cone operators. This presentation is based on joint work with Thomas Krainer and Gerardo Mendoza.

Alexandre Girouard: The Steklov problem is a geometric eigenvalue problem with its spectral parameter at the boundary of a compact Riemannian manifold. In this talk, I will describe a joint work with L. Parnovski (UCL), I. Polterovich (U. Montréal) and D. Sher (U. Michigan) in which we have obtained precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. This has led to a proof that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. Consequences include an upper bound for the multiplicity of eigenvalues on planar domains, as well as a spectral rigidity result for the disk.

Carolyn Gordon: The Dirichlet-to-Neumann operator of a compact Riemannian manifold M with boundary is a linear map $C^\infty(\partial M)\to C^\infty(\partial M)$ that maps the Dirichlet boundary values of each harmonic function f on M to the Neumann boundary values of f. The spectrum of this operator is discrete and is called the Steklov spectrum. We will discuss joint work with Peter Herbrich and David Webb concerning the construction of pairs of Steklov isospectral bounded domains in a fixed noncompact manifold. The Laplacians on these domains are also isospectral for both the Dirichlet and Neumann boundary problems and the exterior domains are isophasal. The latter result is joint with Peter Perry.

Victor Guillemin: In this talk I will discuss some recent joint work with Emily Dryden and Rosa Sena-Dias on equivariant spectral theory. Our results have precedents in work of Donnelly and Bruning-Heintze from the 1980's. Namely in their papers they consider, for a group, G, acting on a compact Riemannian manifold, M, the spectrum of the Laplace operator on spaces of functions which transform under the action of G by a fixed character (or in the case of abelian groups a fixed weight, alpha) of G. The novelty of our approach (which I'll describe in detail in my talk) is to consider, in place of the Laplace operator, its "semi-classical" counterpart, and replace the weights, alpha by "semi-classical" weights alpha/h. This leads to a rather different, and in some ways simpler version of equivariant spectral theory. In particular it enables us to avoid doing analysis on the quotient space, M/G, which can often have rather bad singularities. (This talk is in addition a prequel to the talk of Zuoqin Wang who will discuss some inverse spectral applications of these ideas.)

Stuart Hall: The Kähler-Einstein problem on toric-Kähler manifolds is completely solved due to a result of Wang and Zhu. Unsurprisingly, the existence theorem is not constructive. However, what is particularly attractive about this family of metrics is that they can be considered using a framework developed by Guillemin and Abreu. The geometry of the metrics can be understood by considering the behaviour of convex functions on certain special polytopes. By using the Matsushima theorem and integration-by-parts (both gems of different antiquities) we were able to obtain bounds for the second non-zero eigenvalue of the Laplacian (ignoring multiplicities). We computed these bounds for various low-dimensional examples and compared them with more accurate (sometimes explicit) calculations of the eigenvalue. These results suggest the role that the second eigenvalue might play in determining the geometry of toric Kähler-Einstein metrics. This talk will explain these results and conjectures. This is joint work with Tommy Murphy.

Eveline Legendre: Toric Kähler geometry is a rich playground, where underlying (symplectic) orbifolds correspond to polytopes and (Kähler) metrics to certain convex functions on these polytopes. In an ongoing joint project with Rosa Sena-Dias, we are studying variations of the first eigenvalue corresponding to variation of the Kähler metric on a fixed symplectic toric orbifold. We prove that the first torus invariant eigenvalue is unbounded, extending work of Abreu and Freitas on the 2 sphere. Using the work of Bourguignon-Li-Yau we give an explicit bound for the first eigenvalue of compact toric Kähler manifold in terms of the polytope.

Iosif Polterovich: The classical Hardy-Landau lower bound for the error term in the Gauss circle problem can be viewed as an estimate from below for the remainder in Weyl's law for the eigenvalue counting function on a torus. In the talk we will present an analogous estimate for certain planar domains admitting an appropriate one-parameter family of periodic billiard trajectories. Examples include ellipses and smooth domains of constant width. In higher dimensions, lower bounds on the remainder in Weyl's law are of somewhat different nature, and they will be discussed as well. The talk is based on a joint work with Suresh Eswarathasan and John Toth.

Dorothee Schueth: It is well-known that the spectrum on functions of a Riemannian manifold does not determine the integrals of the individual fourth order curvature invariants $scal^2$, $|ric|^2$, $|R|^2$, which appear as summands in the second heat invariant a_2. We study the analogous question for the sixth order curvature invariants constituting a_3. None of them appears to be determined individually by the spectrum, which can be shown using various examples. In particular, we prove that two isospectral nilmanifolds of Heisenberg type with three-dimensional center are locally isometric if and only if they have the same value of $|\nabla R|$^2. In contrast, any pair of isospectral nilmanifolds of Heisenberg type with centers of dimension greater than three does not differ in any of the sixth order curvature invariants, in spite of local nonisometry. This is joint work with Teresa Arias Marco.

David Sher: We consider the problem of estimating nodal length - that is, length of the zero sets - of Steklov eigenfunctions on a surface with boundary. A conjecture in the spirit of Yau statesthat the nodal length of a Steklov eigenfunction should be bounded above and below by a geometric constant times the associated Steklov eigenvalue. We prove this conjecture under the assumption that the surface with boundary is real analytic. This is joint work with I. Polterovich (Montreal) and J. Toth (McGill).

Craig Sutton: Motivated in part by considerations from quantum mechanics and geometric optics, it is conjectured that the length spectrum of a closed manifold can be recovered from its Laplace spectrum. We demonstrate that the length spectrum of a compact simple Lie group equipped with a bi-invariant metric can be recovered from its Laplace spectrum by computing the singular support of the trace of its associated wave group; that is, we show that the Poisson relation is an equality for such spaces. More generally, we see that the conjecture holds for a split-rank symmetric space M = G/K satisfying any one of the following conditions: (1) M is simply-connected (e.g., a simply-connected Lie group equipped with a bi-invariant metric); (2) the metric on M is the unique up to scaling G-invariant Einstein metric (e.g., a compact semi-simple Lie group equipped with the bi-invariant metric induced by the Killing form), or (3) the universal cover of M is a product of irreducible split-rank factors coming from certain infinite families (e.g., certain infinite families of compact semi-simple Lie groups equipped with a bi-invariant metric).

Alejandro Uribe: We consider Schrödinger operators with complex potentials on Zoll manifolds. We determine the asymptotic pseudospectrum and numerical range of the operator, in clusters around the eigenvalues of the unperturbed laplacian. The proofs involve constructing quasi modes for the operator, as well as for the Zoll laplacian. This is joint work with D. Sher and C. Villegas-Blas.

Zuoqin Wang: By using the equivariant spectral invariants developed by Emily Dryden, Victor Guillemin and Rosa Sena-Dias, we will show how to recover nice T^n-invariant potentials for the semi-classical Schrodinger operators on toric varieties via the equivariant spectrum. The main tool is a generalized version of the classical Legendre transform in the frame of symplectic geometry. This is joint work with Victor Guillemin.