Program for the Year 2020 to 21 | Spectral Geometry in the Clouds

Program of the seminar for the Academic Year 2020 to 2021

August 31	Gerasim Kokarev (University of Leeds)
	Comparison principles, minimal surfaces, and eigenvalue inequalities
	View Abstract I will talk about comparison principles in (spectral) geometry, describe related phenomena and applications to eigenvalue inequalities. In particular, I will tell about new inequalities for Laplace eigenvalues in terms of Berger’s embolic volume. Time permitting, I will also discuss comparison principles and eigenvalue bounds for minimal surfaces.
September 7	Jean Lagacé (University College London)
	Measure theoretic methods in spectral shape optimisation
	View Abstract Shape optimisation is a recurrent theme in spectral geometry: given an operator and some boundary conditions, what is the optimal domain for the associated eigenvalue problem. Traditionally, different boundary conditions and different operators are treated independently. I will present a unifying framework based on measure theory for treating many different optimisation problem at once, and to relate them to one another. As an application, we will see that from the shape optimisation perspective, the Steklov, Laplace, and Neumann problems can be regrouped as part of a larger class, and how to use those methods to look for optimisers and their properties. Based on joint work with Alexandre Girouard (Laval) and Mikhail Karpukhin (Caltech)
September 14	Alix Deleporte (University of Zürich)
	Semiclassical analysis of a problem with a circular well
	View Abstract In joint work with San Vũ Ngọc we study classical and quantum systems with one degree of freedom, for which the energy profile (defined on a surface) reaches its minimum on a closed curve. A typical example is a massive particle moving on a circle and subject to a magnetic field. The quantum version of such problems is, in particular, useful as an effective model for studying magnetic fields in several degrees of freedom, and we exhibit an oscillation phenomenon for small eigenvalues, similar to the "Little-Parks" in supraconductors. In this talk, I will show how to "solve" the classical problem, that is, find good coordinates which simplify it: Then I will discuss the quantum version of this simplification, and notably the origin of these oscillations. This talk is, hopefully, accessible without any particular knowledge of semiclassical analysis.
September 21	Davide Buoso (Università degli Studi del Piemonte Orientale “A. Avogadro”)
	The Bilaplacian with Robin boundary conditions
	View Abstract In this talk we present a new type of boundary conditions for the Bilaplacian, that we will call Robin boundary conditions. This new problem can be thought of as a model for elastically supported plates, and moreover turns out to be closely related to the study of spaces of traces of Sobolev functions. We will start by recalling some known properties of the Robin Laplacian as a motivation for its fourth-order counterpart, then we will introduce the Robin Bilaplacian operator. We will then analyze the dependence of the operator, its eigenvalues, and eigenfunctions on the Robin parameters, with a particular focus on the asymptotics as the parameters tend to infinity. We will conclude by considering the dependence on smooth perturbations of the domain, computing the shape derivatives of the eigenvalues and giving a characterization for critical domains under volume and perimeter constraints. Based on a joint work with J. Kennedy.
September 28	Jeremy Marzuola (University of North Carolina at Chapel Hill)
	A local test for global extrema in the dispersion relation for periodic graphs
	View Abstract With Greg Berkolaiko, Yaiza Canzani and Graham Cox, we consider a family of periodic tight-binding models (which are combinatorial graphs with Laplacian operators that have edge weights parametrized over tori). It will be important that our graphs have the minimal number of links between copies of the fundamental domain. For this family of graphs, we establish a local condition of second derivative type under which the critical points of the dispersion relation can be recognized as global maxima or minima. We will try to give an introduction to periodic graphs, as well as their applications. In addition, we will demonstrate our results with a number of example graphs.
October 5	Richard Laugesen (University of Illinois)
	Two balls maximize the third Neumann eigenvalue in hyperbolic space
	View Abstract The second eigenvalue of the Neumann Laplacian is maximal for a ball under volume normalization, by Weinberger (1956). For simply connected planar domains, Szegö did it two years earlier using conformal mapping. The third eigenvalue is maximal for a disjoint union of two balls having the same size, by Bucur and Henrot (2019). For simply connected planar domains, Girouard, Nadirashvili and Polterovich did it ten years earlier. This talk explores the general approach of these authors in terms of trial functions constructed by even reflection (“folding over a hyperplane”), and provides a new proof of a key step in Bucur and Henrot’s argument. A starring role is played by a homotopy result of Petrides that has the flavor of the Borsuk-Ulam theorem. This new approach yields a two-ball upper bound on the third eigenvalue for domains in hyperbolic space. (Joint work with Pedro Freitas.)
October 12	Sugata Mondal (Tata Institute of Fundamental Research)
	Hot spots conjecture for polygonal domains
	View Abstract The hot spots conjecture says that a second Neumann eigenfunction of a domain in a Euclidean space can not attain its global extrema in the interior of the domain. I shall try to provide a brief survey of the known results on this topic. In particular, I will try to explain a recent result of myself with Chris Judge that provides an affirmative answer to the conjecture for triangular domains in the plane.
October 19	Antoine Henrot (Institut Elie Cartan)
	Isodiametric inequalities for eigenvalues
	View Abstract Among optimization problems for eigenvalues od Euclidean domains, the most common geometric constraints which have been considered are the volume or perimeter constraints. Here in this talk we consider a constraint on the diameter. First, we consider the minimization of the Dirichlet eigenvalues. It is easy to see that the minimizer exists and is a convex domain of constant width. Then we prove that the disk is a local minimizer only for a finite (and explicit) list of indices. At last, we study the limit of the optimal domain for \(\lambda_k\) when \(k\to \infty\). Then, we consider the problem of maximizing the \(k\)-th Steklov eigenvalue among convex domains. Existence of a maximizer is not difficult to prove. Even if the disk is always a critical point, we prove that it is never a maximizer. We also prove that, for the optimal domain, the eigenvalue is always multiple. In both cases, we present some numerical results. This is a joint work with Beni Bogosel, Ilaria Lucardesi, Abdelkader Al Sayed and Florent Nacry.
October 26	Christiane Tretter (Universität Bern)
	Spectral problems on star graphs
	View Abstract In this talk various results for spectral problems on star graphs are presented, e.g. for Stieltjes strings or Dirac operators. Apart from direct and inverse spectral results, we establish an abstract reduction method collapsing the problem on a star graph to a path graph. (joint with N. Rozhenko/V. Pivovarchik and B.M. Brown/H. Langer)
November 2	Stefano Decio (Norwegian University of Science and Technology)
	Nodal sets of Steklov eigenfunctions near the boundary
	View Abstract We will discuss zero sets of Steklov eigenfunctions: we show that there are many zeros near the boundary and improve on previous lower and upper bounds for the Hausdorff measure of the zero set; several questions remain unanswered. Comparisons with the (slightly) better understood case of eigenfunctions of the Laplace-Beltrami operator will also be provided.
November 9	Jeffrey Galkowski (University College London)
	The interior behavior of Steklov eigenfunctions
	View Abstract The Steklov problem consists of studying the eigenvalues and eigenfunctions for the Dirichlet to Neumann map on a compact manifold, \(M\), with boundary. While the high energy behavior of the eigenfunctions on the boundary of \(M\) resembles that of high energy Laplace eigenfunctions, their behavior in the interior of \(M\) is very different. In this talk, we discuss the decay and oscillatory properties of these eigenfunctions in the interior of \(M\).
November 16	Katie Gittins (Durham University)
	Upper bounds for Steklov eigenvalues
	View Abstract The relationship between the Steklov eigenvalues of a Riemannian manifold and the geometry of its boundary has received a great deal of attention in recent years. One way to gain insight into this relationship is to obtain bounds for the Steklov eigenvalues in terms of some of the geometric quantities of the manifold and its boundary. We first recall some known geometric upper bounds for the Steklov eigenvalues of a Riemannian manifold. We then consider the Steklov eigenvalues of a submanifold of Euclidean space and present some geometric upper bounds involving the intersection index of the manifold and that of its boundary. This is based on joint work with Bruno Colbois (Université de Neuchâtel).
November 23	Peter Sarnak (IAS)
	Gap Sets for the Spectra of Cubic Graphs
	View Abstract The spectra of large locally uniform geometries have been studied widely and from different points of view, including in applications. They include Ramanujan Graphs and Buildings, euclidean and hyperbolic spaces and more general locally symmetric spaces. We review some of these briefly highlighting rigidity features. We then focus on the simplest case of finite cubic graphs which prove to be surprisingly rich. As one imposes restrictions on these graphs, planarity, fullerenes, ... their spectra become rigid. Joint work with Alicia Kollar and Fan Wei.
November 30	Édouard Oudet (Université Grenoble Alpes)
	Metric Optimization in Spectral geometry
	View Abstract The results of this talk have been obtained in collaboration with Chiu-Yen Kao and Braxton Osting. The first part of the talk is dedicated to Nash's isometric embedding theorem for surfaces. We recall the impressive results obtained by HEVEA's project for the flat torus (V. Borrelli, F. Lazarus, B. Thibert et al.) and illustrate how spectral formulation may lead to a new intrinsic numerical approach which are not related to Gromov's construction. Following the theoretical results of Fraser and Schoen, we describe in a second part a numerical process to approximate minimal surfaces in the ball. That is surfaces contained in the ball that have zero mean curvature and meet the boundary of the ball orthogonally. For genus \(\gamma=0\) and \(b= 2, \cdots , 9, 12, 15, 20\) boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which have not been observed previously.
December 7	Simon Larson (Caltech)
	Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain
	View Abstract We consider the Dirichlet Laplace operator in a bounded domain with Lipschitz- regular boundary and study the asymptotic behaviour of its eigenvalues. In particular, we prove a two-term asymptotic expansion for the sum of eigenvalues and show that the obtained remainder is optimal. The talk is based on joint work with Rupert Frank.
December 14	Marco Marletta (Cardiff School of Mathematics)
	A Laplace operator with boundary conditions singular at one point
	View Abstract In this talk I will present some work with Rozenblum from 2009. While it has been known for more than half a century that the Laplace operator on a smooth, bounded domain may have essential spectrum if the boundary conditions are suitably chosen, typical choices involved non-local operators. In this talk I will show, with very elementary arguments, that even local boundary conditions, singular even just at a single point - can have a huge impact on the spectrum and eigenfunctions. The example we consider, first proposed by Berry and Dennis. still has empty essential spectrum and compact resolvent. However Weyl’s law fails completely because the spectrum becomes unbounded below. The positive eigenvalues still obey Weyl asymptotics, to leading order; however the (absolute values of the) negative eigenvalues do not obey a power law distribution. I will also make some remarks and ask some questions about nodal domains, which were not addressed in our paper.
January 11	Olaf Post (Humboldt-Universität)
	Operator estimates for wildly perturbed manifolds
	View Abstract We show norm resolvent convergence for the Laplacian on a family of domains with many small holes taken out. The critical parameter here is the capacity of the small holes. If the density of holes is below the critical parameter, then the limit operator is the free Laplacian, in the critical case, an extra potential term appears. The proof relies on an abstract convergence scheme for operators acting in varying Hilbert spaces.
January 18	Pavel Exner (Doppler Institute for Mathematical Physics and Applied Mathematics, Prague)
	Discrete spectrum of Dirichlet Laplacian in spiral-shaped regions
	View Abstract We discuss spectral properties of Laplace operator in spiral-shaped regions with Dirichlet boundary, in particular, their discrete spectrum. As a case study we analyze in detail the region generated by the Archimedean spiral, where in contrast to ‘less curved’ tubular regions the discrete part is empty due to a subtle diﬀerence between the radial and perpendicular widths, and the spectrum is absolutely continuous away from the thresholds. For more general spiral regions the spectral nature depends substantially on whether their coil width is ‘expanding’ or ‘shrinking’ with respect to the angle; the most interesting situation occurs in the case we call asymptotically Archimedean, where the existence of isolated eigenvalues depends on the sign of the leading term in the asymptotics.
January 25	Zeev Rudnick (Tel-Aviv University)
	Robin vs Neumann spectra for planar domains
	View Abstract I will discuss the fluctuations of the gaps between the Robin and Neumann eigenvalues for planar domains. I will present some numerical experimentations which reveal connections with number theory and quantum ergodicity, some new results inspired by these numerical explorations, and several open problems.
February 1	Luigi Provenzano (Sapienza Università di Roma)
	On the spectral representation of the trace spaces of \(H^2\) and the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems
	View Abstract We describe the traces of functions in \(H^2(\Omega)\) on a Lipschitz domain \(\Omega\) in terms of Fourier series associated with the eigenfunctions of multi-parameter Steklov problems, which we introduce for this specific purpose. The corresponding characterization of the trace spaces allows to represent in series the solutions to biharmonic Dirichlet problems. We also present asymptotic properties of the eigenvalues as well as explicit examples. Joint work with Pier Domenico Lamberti (Università degli Studi di Padova).
February 8	Alessandro Carlotto (ETH Zürich)
	Free boundary minimal surfaces with connected boundary and arbitrary genus
	View Abstract Besides their self-evident geometric significance, which can be traced back at least to Courant, free boundary minimal surfaces also naturally arise in partitioning problems for convex bodies, in capillarity problems for fluids and, as has significantly emerged in recent years thanks to work of Fraser and Schoen, in connection to extremal metrics for Steklov eigenvalues for manifolds with boundary (i. e. for eigenvalues of the corresponding Dirichlet-to-Neumann map). The theory has been developed in various interesting directions, yet many fundamental questions remain open. One of the most basic ones can be phrased as follows: does the Euclidean unit ball contain free boundary minimal surfaces of any given topological type? In spite of significant advances, the answer to such a question has proven to be very elusive. I will present some joint work with Giada Franz and Mario Schulz where we answer (in the affirmative) the well-known question whether there exist in \(B^3\) (embedded) free boundary minimal surfaces of genus one and one boundary component. In fact, we prove a more general result: for any g there exists in \(B^3\) an embedded free boundary minimal surface of genus \(g\) and connected boundary. The proof builds on global variational methods, in particular on a suitable equivariant counterpart of the Almgren-Pitts min-max theory, and on a striking application of Simon's lifting lemma.
February 15	Uzy Smilansky (Weizmann Institute)
	Can one hear a matrix? The determination of a real symmetric matrix from spectral information
	View Abstract The question asked in the title is addressed from two points of view: First, it will be shown that providing enough (term to be explained) spectral data, suffices to reconstruct a wide class of matrices. Second, a newly developed trace formula for Hermitian matrices will be shown to be intimately connected to this construction.
February 22	Peter Kuchment (Texas A&M University)
	Spectral shift via lateral variation
	View Abstract Our study is motivated by earlier results about nodal count of Laplacian eigenfunctions on manifolds and graphs that share the same flavor: the nodal count's "deviation" is equal to the Morse index of a certain "energy functional" . In the hindsight, in all these results, the nodal count can be understood as the spectral shift resulting from perturbing the operator in an appropriate way. This brings us to the following general result (joint with G. Berkolaiko): the spectral shift can be recovered as the stability (Morse) index of the eigenvalue with respect to small "lateral" variations of the perturbation.
March 1	Grigori Rozenblum (St. Petersburg State University)
	Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case
	View Abstract We consider self-adjoint operators of the form \(\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A}\) in a domain \(\Omega\subset \mathbb{R}^\mathbf{N}\), where \(\mathfrak{A}\) is an order \(-l=-\mathbf{N}/2\) pseudodifferential operator in \(\Omega\) and \(P\) is a signed Borel measure with compact support in \(\Omega\). Measure \(P\) may contain singular component. For a wide class of measures we establish eigenvalue estimates for operator \(\mathbf{T}_{P,\mathfrak{A}}\). In case of measure \(P\) being absolutely continuous with respect to the Hausdorff measure on a Lipschitz surface of an arbitrary dimension, we find the eigenvalue asymptotics. The order of eigenvalue estimates and asymptotics does not depend on dimensional characteristics of the measure, in particular, on the dimension of the surface supporting the measure. The typical example is the spectral problem \(\lambda (1-\Delta)^{N/2}u =P u\).
March 8	Graham Cox (Memorial University)
	Nodal deficiency via equipartitions and Dirichlet-to-Neumann maps
	View Abstract Courant's nodal domain theorem, which says the \(n\)th Laplacian eigenfunction has at most \(n\) nodal domains, is almost always a strict inequality. The extent to which it fails to be sharp is measured by the nodal deficiency. Despite much study, this quantity is still not very well understood except in highly symmetric cases. However, in the last decade two general formulas for the nodal deficiency were established. The first was given in 2012 by Berkolaiko, Kuchment and Smilansky, using an energy functional defined on the space of equipartitions of the domain. More recently, with Jones and Marzuola, I obtained a formula for the nodal deficiency in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. After reviewing both of these results, I will describe new work (with Gregory Berkolaiko, Yaiza Canzani and Jeremy Marzuola) that demonstrates a direct connection between these seemingly different approaches to nodal deficiency. Among other things, it gives a method for using the Dirichlet-to-Neumann map to calculate eigenfunctions for the Hessian of the equipartition energy, and provides insight into the theory of spectral minimal partitions.
March 15	Nilima Nigam (Simon Fraser University)
	The approximation of Laplace Steklov eigenvalues
	View Abstract In this talk we present a high-accuracy discretization strategy for computing the Steklov eigenpairs of the Laplacian. We'll also present recent results on the use of this method to investigate questions arising in spectral optimization and the spectra of polygons.
March 21-22	Mini Conference: Young researchers in spectral geometry II
March 21 8AM PDT 11AM EDT 3PM GMT 4 PM CET	Mickael Nahon (Université Savoie Mont Blanc)
	Degenerate free discontinuity problems and quantitative spectral inequalities
	View Abstract We study a semilinear eigenvalue problem involving the Laplace operator with Robin boundary conditions among sets of fixed measure in \(\mathbb{R}^n\). In particular, consider a solid \(D\) with a constant volumetric heat source and a thin layer of insulator on its surface, and let \(T(D)\) be the mean temperature of \(D\) at equilibrium. We will present a proof of \(T(D)\leq T(D^)-C.d(D,D^)^2\), where \(d\) is the \(L^1\) distance between sets up to translation and \(D^*\) is the ball with same measure as \(D\). This will be obtained through a shape optimisation problem involving both free discontinuity and free boundary phenomena.
March 21 9AM PDT 12AM EDT 4PM GMT 5 PM CET	Hanna Kim (University of Illinois)
	Maximization of the second Laplacian eigenvalue on the sphere
	View Abstract The second nonzero eigenvalue of the Laplacian on \(S^{m}\) becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. This talk builds on recent developments on the hyperbolic center of mass to simplify the proof, yielding also the analogous result in higher dimensions.
March 21 10AM PDT 1PM EDT 5PM GMT 6 PM CET	Samuel Perez-Ayala (University of Notre Dame)
	Extremal Eigenvalues of the Conformal Laplacian
	View Abstract I will report on joint work with Mathew J. Gursky in which we consider the problem of extremizing eigenvalues of the conformal laplacian in fixed conformal classes. I will explain the connection of these extremal metrics to constant curvature metrics (Yamabe metrics), to the existence of harmonic maps into spheres, and to nodal solutions of a Yamabe type equation (first noticed by Ammann-Humbert).
March 22 8AM PDT 11AM EDT 3PM GMT 4 PM CET	Germain Gendron (Université de Nantes)
	Uniqueness and stability for an inverse Steklov problem
	View Abstract In this talk, we present results for an inverse Steklov problem for a particular class of 2-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance. Then, we study the continuous dependence of the warping function defining the warped product with respect to the Steklov spectrum.
March 22 9AM PDT 12AM EDT 4PM GMT 5 PM CET	Laura Monk (IRMA Strasbourg)
	Geometry and spectrum of random hyperbolic surfaces
	View Abstract The main aim of this talk is to present geometric and spectral properties of typical hyperbolic surfaces. More precisely, I will: introduce a probabilistic model, first studied by Mirzakhani, which is a natural and convenient way to sample random hyperbolic surfaces describe the geometric properties of these random surfaces explain how one can deduce from this geometric information estimates on the number of eigenvalues of the Laplacian in an interval \([a,b]\), using the Selberg trace formula.
March 22 10AM PDT 1PM EDT 5PM GMT 6 PM CET	Zhichao Wang (University of Toronto)
	Conformal upper bounds for the volume spectrum
	View Abstract In this talk, we prove upper bounds for the volume spectrum of a Riemannian manifold that depend only on the volume, dimension, and a conformal invariant.
March 29	Mariel Saez (Pontificia Universidad Católica de Chile)
	Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds
	View Abstract In this talk I will discuss bounds for the first eigenvalue of the Paneitz operator \(P\) and its associated third-order boundary operator \(B^3\) on four-manifolds. We restrict to orientable, simply connected, locally confomally flat manifolds that have at most two umbilic boundary components. The proof is based on showing that under the hypotheses of the main theorems, the considered manifolds are confomally equivalent to canonical models. The fact that \(P\) and \(B^3\) are conformal in four dimensions is key in the proof. This is joint work with Maria del Mar Gonzalez
April 12	Svitlana Mayboroda (Université Paris Sud)
	The landscape law for the integrated density of states
	View Abstract We establish non-asymptotic estimates from above and below on the integrated density of states of the Schrödinger operator \(L=-\Delta+V\), using a counting function for the minima of the localization landscape, a solution to the equation \(Lu=1\).
April 19	Bernard Helffer (Université Paris Sud)
	Semi-classical edge states for the Robin Laplacian (after Helffer-Kachmar)
	View Abstract Motivated by the study of high energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative eigenvalues, and we prove that the corresponding eigenfunctions decay away from the boundary, for all dimensions.
April 26	Karl-Mikael Perfekt (Université Paris Sud)
	Infinitely many embedded eigenvalues for the Neumann-Poincaré operator in 3D
	View Abstract I will discuss the spectral theory of the Neumann-Poincaré operator for 3D domains with rotationally symmetric singularities, which is directly related to the plasmonic eigenvalue problem for such domains. I will then describe the construction of some special domains for which the problem features infinitely many eigenvalues embedded in the essential/continuous spectrum. Several questions and open problems will be stated. Based on joint papers with Johan Helsing and with Wei Li and Stephen Shipman.
May 3	Jared Wunsch (Northwestern University)
	Semiclassical analysis and the convergence of the finite element method
	View Abstract An important problem in numerical analysis is the solution of the Helmholtz equation in exterior domains, in variable media; this models the scattering of time-harmonic waves. The Finite Element Method (FEM) is a flexible and powerful tool for obtaining numerical solutions, but difficulties are known to arise in obtaining convergence estimates for FEM that are uniform as the frequency of waves tends to infinity. I will describe some recent joint work with David Lafontaine and Euan Spence that yields new convergence results for the FEM which are uniform in the frequency parameter. The essential new tools come from semiclassical microlocal analysis and the use of the functional calculus. Another ingredient is a slightly surprising new resolvent estimate.
May 10	Ram Band (Technion)
	Neumann domains on manifolds and graphs
	View Abstract The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains (you may guess the reason for this name, and it would also be mentioned in the talk ;) We present results for Neumann domains on manifolds and on graphs - their count, geometric properties and spectral positions. The Neumann domain results are compared to those of the nodal domain study. The talk is based on joint works with Lior Alon, Michael Bersudsky, Graham Cox, Sebastian Egger, David Fajman and Alexander Taylor.
May 17	Alexandre Girouard (Université Laval)
	Free boundary minimal surfaces and large Stekov eigenvalues
	View Abstract A free boundary minimal surface (FBMS) in the unit ball is a minimal surface \(\Omega\subset\mathbb{B}\subset\mathbb{R}^3\) which satisfies one of the two following equivalent conditions: \(\Omega\) meets the boundary \(\partial\mathbb{B}\) orthogonally, The coordinate functions \(x_i:\Omega\to\mathbb{R}\) are Steklov eigenfunctions with eigenvalue \(\sigma=1\). The study of FBMS is witnessing a renaissance thanks to the fundamental work of Fraser and Schoen, who discovered that isoperimetric optimizers for the first nonzero Steklov eigenvalue \(\sigma_1(\Omega)\) of surfaces lead to the existence of FBMS with prescribed topology. For surfaces of genus 0, they proved existence of maximizers for the perimeter-normalized \(\sigma_1(\Omega)\) and thereby obtained several new FBMS. In this talk I will show how this link can be used to obtain a sequence of FBMS \(\Omega^n\subset\mathbb{B}\) such that \(\text{area}(\Omega^n)\rightarrow 4\pi\) as \(n\to\infty\). This is based on a construction of domains in the sphere \(\mathbb{S}^2\) with perimeter-normalized \(\sigma_1(\Omega)\) converging to \(8\pi\). These domains are obtained by removing small disks from the sphere, in the spirit of homogenization theory. I will also dicuss a conjecture by Fraser and Li, stating that \(\sigma_1(\Omega)=1\) for each FBMS \(\Omega\subset\mathbb{B}\). More precisely, I will show that the conjecture is true for some FBMS which are invariant under the action of the symmetry group of some platonic solids. This talk is based on joint work with Jean Lagacé (UCL).
May 24	Genqian Liu (Beijing Institute of Technology)
	Geometric invariants of spectrum of the Navier-Lame operator
	View Abstract In this talk, we review the asymptotic expansions of the heat traces for various operators including the Laplace operator, the poly-Laplace operator, the Maxwell operator, the Stokes operator, etc. Then for the elastic Navier-Lame operator (a non-Laplace type operator) on a compact connected Riemannian \(n\)-manifold \(M\) with smooth boundary, we explicitly obtain the first two coefficients of the asymptotic expansion of the heat trace for Navier-Lame operator with Dirichlet and Neumann boundary conditions. These two coefficients provide precise information for the volume of the elastic body \(M\) and the surface area of the boundary in terms of the spectrum of the Navier-Lame operator. This gives an answer to an interesting and open problem mentioned by Avramidi.
May 31	Maziej Zworski (Berkeley)
	Spectral theory of internal waves in fluids
	View Abstract The connection between the formation of internal waves in fluids, spectral theory and the dynamics of homeomorphisms of the circle was investigated by oceanographers in the 90s and resulted in novel experimental observations (Maas et al, 1997). The specific homeomorphism is given by a chess billiard and has been considered by many authors (John 1941, Arnold 1957, Ralston 1973, ... , Lenci et al 2021). The relation between the nonlinear dynamics of this homeomorphism and linearized internal waves provides a striking example of classical/quantum correspondence (in a classical and surprising setting of fluids!). Using a model of tori and of zeroth order pseudodifferential operators, it has been a subject of recent research, first by Colin de Verdière-Saint Raymond 2020 and then by Dyatlov, Galkowski, Wang and the speaker. In this talk I will review those results and present new work, with Dyatlov and Wang, on the more physically relevant boundary value problem.
June 7	Mikhail Karpukhin (California Institute of Technology)
	Stability of isoperimetric eigenvalue inequalities
	View Abstract Stability questions for sharp inequalities are important problems in analysis. Recently, these questions have been investigated for the first eigenvalue of the Laplacian on Euclidean domains. Optimal stability estimates for Faber-Krahn and Szego-Weinberger inequalities were obtained by Brasco-De Philippis-Velichkov and Nadirashvili, Brasco-Pratelli respectively. In the present talk we first consider the stability of another fundamental inequality in spectral geometry: Hersch inequality for the first eigenvalue on the 2-dimensional sphere. We then present generalizations to other surfaces and the related problems from harmonic maps and minimal surfaces. Finally, if time permits, potential applications to Steklov eigenvalue problem will be discussed. Based on the joint work with M. Nahon, I. Polterovich and D. Stern.
June 14	Javier Gómez-Serrano (University of Barcelona)
	Computer-assisted proofs and counterexamples in spectral geometry
	View Abstract In this talk I will explain how to construct counterexamples for two problems in spectral geometry. The main novelty is that parts of the proofs will be done via a rigorous computer-assisted proof. In the first part of the talk, I will explain how to prove that a triangle is not determined by its first, second and fourth (Dirichlet) eigenvalues, solving a conjecture by Antunes and Freitas. In the second part I will construct a planar domain with 6 holes for which the nodal line is closed and does not touch the boundary. In particular, this domain does not satisfy Payne's nodal line conjecture. This gives a partial answer on a question posed by Hoffmann-Ostenhof, Hoffmann-Ostenhof and Nadirashvili asking what should be the minimal number of holes of such domains. The results are joint work with Joel Dahne, Kimberly Hou and Gerard Orriols.
June 21	Paolo Luzzini (EPFL)
	On the spectral asymptotics for the buckling problem
	View Abstract Since the seminal works of Hermann Weyl at the beginning of the 20th century, several authors have investigated the spectral asymptotics of partial differential operators. Following this tradition, in this talk I will first present a recent result on a new proof of Weyl's law for the buckling eigenvalues requiring minimal assumptions on the domain. The proof relies on asymptotically sharp lower and upper bounds that we develop for Riesz means. Moreover, we compute the second term in Weyl's law in the case of balls and bounded intervals. This, together with some formal considerations, leads us to state a conjecture for the second term in general domains. The talk is based on a joint work with Davide Buoso (UPO), Luigi Provenzano (Sapienza Università di Roma), and Joachim Stubbe (EPFL).
June 28	Ari Laptev (Imperial College London)
	Spectral inequalities for Jacobi matrices following Hundertmark and Simon
	View Abstract We shall proof of a Lieb-Thirring type inequality for Jacobi matrices originally conjectured by Hundertmark and Simon. In particular, we show that the estimate on the sum of eigenvalues does not depend on the off-diagonal terms as long as they are smaller than their asymptotic value. An interesting feature of the proof is that it employs a technique originally used by Hundertmark-Laptev-Weidl concerning sums of singular values for compact operators. This is a joint work with M.Loss and L.Schimmer.
July 5	Steve Zelditch (Northwestern University)
	Eigenfunction restriction problems
	View Abstract My talk is about Fourier expansions of restrictions of eigenfunctions on a Riemannian manifold \(M\) to a submanifold \(H\), in terms of eigenfunctions of \(H\). This problem originated in automorphic forms in work of Hecke, Maass, Selberg, Roelcke, Rankin and others and further developed by Bruggemann, Kuznecov, Hejhal, Good, Wolpert and Marshall on hyperbolic surfaces. More recently, there are many articles studying \(L^2\) norms of restrictions or their integrals over \(H\) against a fixed function. Recently, Wyman, Xi and I have been studying general Fourier coefficients of restrictions with constraints on the ratio of the M-eigenvalue and the \(H\)-eigenvalue. Extremes are restrictions of Gaussian beams along a closed geodesic to the geodesic (which have just one non-zero Fourier coefficient of maximal size) or restrictions of ergodic eigenfunctions of negatively curved surfaces to a close geodesic, where one expects all Fourier coefficients in the allowed range to be of the same small size. My talk will review some of the classical results for motivation and then focus on recent (in progress) results at the edge of the Fourier spectrum when \(H\) is totally geodesic.
July 12	Derek Kielty (University of Illinois Urbana-Champaign)
	Degeneration of the spectral gap with negative Robin parameter
	View Abstract The spectral gap of the Neumann and Dirichlet Laplacians are each known to have a sharp positive lower bound among convex domains of a given diameter. Between these cases, for each positive value of the Robin parameter an analogous sharp lower bound on the spectral gap is conjectured. In this talk we show the extension of this conjecture to negative Robin parameters fails by proving that the spectral gap of double cone domains are exponentially small, for each fixed parameter value.
July 19	Yannick Sire (Johns Hopkins University)
	Eigenfunction and cluster estimates for Schrodinger operators on manifolds
	View Abstract I will describe recent results on several estimates for eigenfunctions, cluster and quasi-modes of schrodinger operators on manifolds with critically singular potentials. Such results lead to space-time estimates for wave equations with potential and are discrete versions of restriction theorems in harmonic analysis. I will also state open problems.
July 26	Victor Ivrii (U. Toronto)
	Pointwise Spectral Asymptotics out of the Diagonal
	View Abstract We establish semiclassical asymptotics or estimates for the Schwartz kernel \(e_h(x,y;\tau)\) of spectral projector for a second order elliptic operator on a manifold with a boundary. While such asymptotics for its restriction to the diagonal \(e_h(x,x,\tau)\) and, especially, for its trace \(N_h(\tau)= \int e_h(x,x,\tau)\,dx\) are well-known, the out-of-diagonal asymptotics are much less explored. Our main tools: improved successive approximations and geometric optics. Our results would also lead to \emph{classical} asymptotics of \(e_h(x,y,\tau)\) for fixed \(h\) (say, \(h=1\)) and \(\tau\to \infty\).