Program for the Year 2021 to 22 | Spectral Geometry in the Clouds

Program of the seminar for the Academic Year 2021 to 2022

August 23	Wencai Liu (Texas A&M University)
	Small denominators and large numerators of quasiperiodic Schrödinger operators
	View Abstract We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schr\"odinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let \((H_{\lambda,\alpha,\theta}u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos2\pi(\theta+n\alpha)u(n)\) be the almost Mathieu operator on \(\ell^2(\mathbb{Z})\), where \(\lambda, \alpha, \theta\in \mathbb{R}\). Let \( \beta(\alpha)=\limsup_{k\rightarrow \infty}-\frac{\ln \|\|k\alpha\|\|_{\mathbb{R}/\mathbb{Z}}}{\|k\|}\). We prove that for any \(\theta\) with \(2\theta\in \alpha \mathbb{Z}+\mathbb{Z}\), \(H_{\lambda,\alpha,\theta}\) satisfies Anderson localization if \(\|\lambda\|>e^{2\beta(\alpha)}\). This confirms a conjecture of Avila and Jitomirskaya [The Ten Martini Problem. Ann. of Math. (2) 170 (2009), no. 1, 303—342] and a particular case of a conjecture of Jitomirskaya [Almost everything about the almost Mathieu operator. II. XIth International Congress of Mathematical Physics (Paris, 1994), 373—382, Int. Press, Cambridge, MA, 1995].
August 30	Jürgen Jost (Max Planck Institute for Mathematics — Leipzig)
	Spectra of graphs and hypergraphs
	View Abstract The spectral theory of the Laplace operator on graphs offers many analogies with that of Riemannian manifolds, like Cheeger type inequalities, but also shows some different phenomena. For hypergraphs, a main step consists in the definition of a Laplace operator that can also offer such analogies. Lovasz extensions of Rayleigh quotients can uncover some deeper reasons behind such analogies.
September 6	Nalini Anantharaman (Université de Strasbourg)
	Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization
	View Abstract The aim of the talk is to discuss the phenomenon of "quantum ergodicity" of eigenfunctions on large expander graphs. We will start with the work of Anantharaman-Le Masson and Anantharaman-Sabri concerning discrete graphs, but the goal is to prove similar results for large quantum graphs. Last year, we proved the same statement for quantum graphs that had been obtained for discrete graphs : if a sequence of finite quantum expander graphs "converges" to an infinite tree, and the latter has purely absolutely continuous spectrum in some given interval \(I\), then "most" of the eigenfunctions of the finite graphs with frequency in \(I\) are delocalized in the sense of quantum ergodicity (which is a rather weak notion of delocalization). Note that the size of the graph goes to infinity, not the frequency, and that we are now dealing with unbounded Schrödinger operators, which modifies a lot the arguments, although the phenomenon is the same at the end. This is joint work with M. Ingremeau, M. Sabri, B. Winn. After stating the result, the talk will mostly be dedicated to the definition of Benjamini-Schramm convergence (which is new for quantum graphs) and to examples where the result applies.
September 13	Mini Conference: Young researchers in spectral geometry III
8:30AM PDT 11:30AM EDT 4:30PM BST 5:30 PM CEST	Vladimir Medvedev (Higher School of Economics)
	On the index of the critical Moebius band in the 4-ball
	View Abstract In this talk I will show that the Morse index of the critical Moebius band in the 4-dimensional Euclidean ball equals 5. This result makes use of the quartic Hopf differential technique and a comparison theorem between the index of a free boundary minimal surface in the Euclidean ball and its spectral index. The latter also enables us to reprove a well-known result that the index of the critical catenoid in the 3-ball equals 4. These results are obtained in my paper in progress.
9:10AM PDT 12:10PM EDT 5:10PM BST 6:10 PM CEST	Marco Michetti (Institut Elie Cartan de Lorraine and Université de Lorraine)
	A comparison between Neumann and Steklov eigenvalues
	View Abstract In this talk we present a comparison between the normalized first (non-trivial) Neumann eigenvalue \(\|\Omega\| \mu_1(\Omega)\) for a Lipschitz open set \(\Omega\) in the plane, and the normalized first (non-trivial) Steklov eigenvalue \(P(\Omega) \sigma_1 (\Omega)\). More precisely, we study the ratio \(F (\Omega) := \|\Omega\| \mu_1 (\Omega)/P (\Omega) \sigma_1(\Omega)\). We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets \(\Omega\). Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. In the last part of the talk we present the corresponding Blaschke-Santaló diagrams \((x, y) = (\|\Omega\|\mu_1 (\Omega), P (\Omega) \sigma_1 (\Omega))\) and we state some open problems. This talk is based on a joint work with Antoine Henrot.
9:50AM PDT 12:50PM EDT 5:50PM BST 6:50 PM CEST	Will Hide (Durham University)
	Spectral gaps for random hyperbolic surfaces with cusps
	View Abstract We shall study the discrete spectrum of the Laplacian on random non-compact finite-area hyperbolic surfaces, focusing on the size of the first non-zero eigenvalue i.e. the spectral gap. We shall introduce a model for random surfaces, arising from the Weil-Petersson metric on moduli space. Then we shall discuss some recent results in this model for compact surfaces and their extension to the non-compact case. In particular, we prove the existence of a positive uniform spectral gap of explicit size for random large genus non-compact surfaces.
September 20	Martin Vogel (Université de Strasbourg)
	Spectral asymptotics of noisy non-selfadjoint operators
	View Abstract The spectral theory of non-selfadjoint operators is an old and highly developed subject. Yet it still poses many new challenges crucial for the understanding of modern problems such as scattering systems, open or damped quantum systems, the analysis of the stability of solutions to non-linear PDEs, and many more. The lack of powerful tools readily available for their selfadjoint counterparts, such a general spectral theorem or variational methods, makes the analysis of the spectra of non-selfadjoint operators a subtle and highly varied subject. One fundamental issue of non-selfadjoint operators is their intrinsic sensitivity to perturbations, indeed even small perturbations can change the spectrum dramatically. This spectral instability, also called pseudospectral effect, was initially considered a drawback as it can be at the origin of severe numerical errors. However, recent works in semiclassical analysis and random matrix theory have shown that this pseudospectral effect also leads to new and beautiful results concerning the spectral distribution and eigenvector localization of non-selfadjoint operators with small random perturbations. In this talk, I will discuss recent results and some fundamental techniques involved in the analysis. The talk is partly based on joint work with Anirban Basak, Stéphane Nonnenmacher, Johannes Sjöstrand and Ofer Zeitouni.
September 27	Gabriel Rivière (Université de Nantes)
	Poincaré series and linking of Legendrian knots
	View Abstract On a compact surface of variable negative curvature, I will explain that the Poincaré series associated to the geodesic arcs joining two given points has a meromorphic continuation to the whole complex plane. This is achieved by using the spectral properties of the geodesic flow. Moreover, the value of Poincaré series value at 0 is rationnal in that case and it can be expressed in terms of the genus of the surface by interpreting it in terms of the linking of two Legendrian knots. If time permits, I will explain how this result extends when one considers geodesic arcs orthogonal to two fixed closed geodesics. This is a joint work with N.V. Dang.
October 4	Philippe Charron (Technion)
	Pleijel's theorem for Schrödinger operators
	View Abstract We will discuss some recent results regarding the number of nodal domains of Laplace and Schrödinger operators. Improving on Courant's seminal work, Pleijel's original proof in 1956 was only for domains in \(\mathbb{R}^2\) with Dirichlet boundary conditions, but it was later generalized to manifolds (Peetre and Bérard-Meyer) with Dirichlet boundary conditions, then to planar domains with Neumann Boundary conditions (Polterovich, Léna), but also to the quantum harmonic oscillator (C.) and to Schrödinger operators with radial potentials (C. - Helffer - Hoffmann-Ostenhof). In this recent work, we proved Pleijel's asymptotic upper bound for a much larger class of Schrödinger operators which are not necessarily radial. In this talk, I will explain the problems that arise from studying Schrödinger operators as well as the successive improvements in the methods that led to the current results.
October 18	Daniel Stern (University of Chicago)
	Steklov-maximizing metrics on surfaces with many boundary components
	View Abstract Just over a decade ago, Fraser and Schoen initiated the study of metrics maximizing the first Steklov eigenvalue among all metrics of fixed boundary length on a given surface with boundary. Drawing inspiration from the maximization problem for Laplace eigenvalues on closed surfaces--where maximizing metrics are induced by minimal immersions into spheres — they showed that Steklov-maximizing metrics are induced by free boundary minimal immersions into Euclidean balls, and laid the groundwork for an existence theory (recently completed by important work of Matthiesen-Petrides). In this talk, I'll describe joint work with Mikhail Karpukhin, characterizing the limiting behavior of these metrics on surfaces of fixed genus \(g\) and \(k\) boundary components as \(k\) becomes large. In particular, I'll explain why the associated free boundary minimal surfaces converge to the closed minimal surface of genus \(g\) in the sphere given by maximizing the first Laplace eigenvalue, with areas converging at a rate of \((\log k)/k\).
October 25	Valentin Blomer (Universität Bonn)
	Eigenvalue statistics of flat tori
	View Abstract The Berry Tabor conjecture predicts that the local statistics of eigenvalues of a regular system is Poissonian, at least in generic cases. In this talk, I consider the special case of flat tori which has the attractive feature that arithmetic tools become available. I will explain some ideas and methods from analytic number theory that shed light on this question, in particular with respect to small gaps, large gaps and triple correlation. This covers joint papers with Aistleitner, Bourgain, Radziwill, Rudnick.
November 1	Jonathan Rohleder (Stockholm University)
	A new approach to the hot spots conjecture
	View Abstract It is a conjecture going back to J. Rauch (1974) that the hottest and coldest spots in an insulated homogeneous medium such as an insulated plate of metal should converge to the boundary, for "most" initial heat distributions, as time tends to infinity. This so-called hot spots conjecture can be phrased alternatively as follows: the eigenfunction(s) corresponding to the first non-zero eigenvalue of the Neumann Laplacian on a Euclidean domain should take its maximum and minimum on the boundary only. This has been proven to be false for certain domains with holes, but it was shown to hold for several classes of simply connected or convex planar domains. One of the most recent advances is the proof for all triangles given by Judge and Mondal (Annals of Math. 2020). The conjecture remains open in general for simply connected or at least convex domains. In this talk we provide a new approach to the conjecture. It is based on a non-standard variational principle for the eigenvalues of the Neumann and Dirichlet Laplacians.
November 8	Sabine Boegli (Durham University)
	On the discrete eigenvalues of Schrödinger operators with complex potentials
	View Abstract In this talk I shall present constructions of Schrödinger operators with complex-valued potentials whose spectra exhibit interesting properties. One example shows that for sufficiently large \(p\), namely \(p>(d+1)/2\) where \(d\) is the dimension, the discrete eigenvalues need not be bounded by the \(L^p\) norm of the potential. This is a counterexample to the Laptev—Safronov conjecture (Comm. Math. Phys. 2009). Another construction proves optimality (in some sense) of generalisations of Lieb—Thirring inequalities to the non-selfadjoint case - thus giving us information about the accumulation rate of the discrete eigenvalues to the essential spectrum. This talk is based on joint works with Jean-Claude Cuenin and Frantisek Stampach.
November 15	Ben Sharp (University of Leeds)
	Łojasiewicz-type inequalities for the \(H\)-functional near simple bubble trees
	View Abstract The \(H\)-functional \(E\) is a natural variant of the Dirichlet energy along maps \(u\) from a closed surface \(S\) into \(\mathbb{R}^3\). Critical points of \(E\) include conformal parameterisations of constant mean curvature surfaces in \(\mathbb{R}^3\). The functional itself is unbounded from above and below on \(H^1(S,\mathbb{R}^3)\), but all critical points have \(H\)-energy \(E\) at least \(4\pi/3\), with equality attained if and only if we are parametrising a round sphere (so \(S\) itself must be a sphere) - this is the classical isoperimetric inequality. Here we will address the simple question: can one approach the natural lower energy bound by critical points along fixed surfaces of higher genus? In fact we prove more subtle quantitative estimates for any (almost-)critical point whose energy is close to \(4\pi/3\). Standard theory tells us that a sequence of (almost-)critical points on a fixed torus \(T\), whose energy approaches \(4\pi/3\), must bubble-converge to a sphere: there is a shrinking disc on the torus that gets mapped to a larger and larger region of the round sphere, and away from the disc our maps converge to a constant. Thus the limiting object is really a map from a sphere to \(\mathbb{R}^3\), and the challenge is to compare maps from a torus with the limiting map (i.e. a change of topology in the limit). In particular we can prove a gap theorem for the lowest energy level on a fixed surface and estimate the rates at which bubbling maps u are becoming spherical in terms of the size of \(dE[u]\) - these are commonly referred to as Łojasiewicz-type estimates. This is a joint work with Andrea Malchiodi (SNS Pisa) and Melanie Rupflin (Oxford).
November 22	Michael Magee (Durham University)
	The maximal spectral gap of a hyperbolic surface
	View Abstract A hyperbolic surface is a surface with metric of constant curvature \(-1\). The spectral gap between the first two eigenvalues of the Laplacian on a closed hyperbolic surface contains a good deal of information about the surface, including its connectivity, dynamical properties of its geodesic flow, and error terms in geodesic counting problems. For arithmetic hyperbolic surfaces the spectral gap is also the subject of one of the biggest open problems in automorphic forms: Selberg's eigenvalue conjecture. It was an open problem from the 1970s whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and spectral gap tending to \(1/4\). (The value \(1/4\) here is the asymptotically optimal one.) Recently we proved that this is indeed possible. I'll discuss the very interesting background of this problem in detail as well as some ideas of the proof. This is joint work with Will Hide.
November 29	David Sher (DePaul University)
	Nodal counts for the Dirichlet-to-Neumann operator
	View Abstract Nodal sets of Steklov eigenfunctions on manifolds with boundary have been extensively studied in recent years. Somewhat less well understood are the nodal sets of their restrictions to the boundary, that is, the eigenfunctions of the Dirichlet-to-Neumann operator. In particular, little is known about nodal counts. In this talk we explore this problem and prove an asymptotic version of Courant's nodal domain theorem for Dirichlet-to-Neumann eigenfunctions. This is joint work with Asma Hassannezhad (Bristol).
December 6	Jean-Claude Cuenin (Loughborough University)
	Schrödinger operators with complex potentials: Beyond the Laptev-Safronov conjecture
	View Abstract I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. This talk can be seen as a continuation of the recent talk by Sabine Bögli (Durham) in the same seminar series, where a counterexample to the Laptev-Safronov conjecture was presented. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. Then I will present new results that show that in some cases one can go beyond the threshold of the counterexample.
December 13	Yann Chaubet (Université Paris-Saclay)
	Closed geodesics with prescribed intersection numbers
	View Abstract On a closed negatively curved surface, Margulis gave the asymptotic growth of the number of closed geodesics of bounded length, when the bound goes to infinity. In this talk, I will present a similar asymptotic result for closed geodesics for which certain intersection numbers — with a given family of pairwise disjoint simple closed geodesics — are prescribed. This result is obtained by introducing a dynamical scattering operator related to the surface (with boundary) obtained by cutting our original surface along the simple curves, and by proving a trace formula.
January 17	Oleksiy Klurman (University of Bristol)
	Boundary-adapted arithmetic random waves and spectral semi-correlations
	View Abstract In this talk, I will discuss recent progress on a class of "tricky" problems concerning additive properties of integral points belonging to the circles. I will focus on spectral applications testing M. Berry's ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions. I will discuss the asymptotic analysis of the nodal length slightly above the Planck scale. This is based on joint works with V. Cammarota and I. Wigman, and A. Sartori.
January 24	Gregory Berkolaiko (Texas A&M University)
	Towards Morse theory of dispersion relations
	View Abstract The question of optimizing an eigenvalue of a family of self-adjoint operators that depends on a set of parameters arises in diverse areas of mathematical physics. Among the particular motivations for this talk are the Floquet-Bloch decomposition of the Schroedinger operator on a periodic structure, nodal count statistics of eigenfunctions of quantum graphs, and the minimal spectral partitions of domains and graphs. In each of these problems one seeks to identify and/or count the critical points of the eigenvalue with a given label (say, the third lowest) over the parameter space which is often known and simple, such as a torus. Classical Morse theory is a set of tools connecting the number of critical points of a smooth function on a manifold to the topological invariants of this manifold. However, the eigenvalues are not smooth due to presence of eigenvalue multiplicities or "diabolical points". We rectify this problem for eigenvalues of generic families of finite-dimensional operators. The correct "Morse indices" of the problematic diabolical points turn out to be universal: they depend only on the total multiplicity at the diabolical point and on the relative position of the eigenvalue of interest in the eigenvalue group. Based on a joint work with I.Zelenko.
January 31	Alessandro Savo (Università di Roma, La Sapienza)
	Isoperimetric inequalities for the lowest Aharonov-Bohm eigenvalue of the Neumann and Steklov problems
	View Abstract We discuss isoperimetric inequalities for the magnetic Laplacian on a bounded domain \(\Omega\) endowed with an Aharonov-Bohm potential \(A\) with pole at a fixed point \(x_0\in\Omega\). Since \(A\) is harmonic on \(\Omega\setminus\{x_0\}\), the magnetic field vanishes; the spectrum for the Neumann condition (or for the Steklov problem) reduces to that of the usual non-magnetic Laplacian, but only when the flux of the potential \(A\) around the pole is an integer. When the flux is not an integer the lowest eigenvalue is actually positive, and the scope of the talk is to show how to generalize the classical inequalities of Szëgo-Weinberger, Brock and Weinstock to the lowest eigenvalue of this particular magnetic operator, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric (which include the spherical and the hyperbolic case).
February 7	Francesco Ferraresso (Cardiff University)
	Neumann and intermediate biharmonic eigenvalue problems on sigularly perturbed domains
	View Abstract Domain perturbation theory for the eigenvalues of the Laplace operator on families of bounded, Lipschitz domains of \(\mathbb{R}^N\) is nowadays a well-understood, yet complicated subject. For the biharmonic operator, the situation is more involved, mainly due to two additional hurdles: 1) intermediate and Neumann boundary conditions are very sensitive to the variation of the curvature of the boundary; 2) standard techniques, such as the separation of variables, are not available. After a review of the main results and counterexamples for the Laplace operator and the biharmonic operator, I will focus on three specific singular perturbations where spectral continuity fails: the dumbbell domain (Neumann b.c); a Lipschitz domain, whose boundary is locally defined as the graph of a fast oscillating smooth function (Intermediate b.c.); thin annuli in \(\mathbb{R}^2\) (Neumann b.c). A particularly striking result is the following. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^2\) with smooth boundary \(\partial \Omega\), and let \(\omega_h\) be the set of points in \(\Omega\) whose distance from the boundary is smaller than \(h\). Then the eigenvalues of the biharmonic operator on \(\omega_h\) with Neumann boundary conditions do not converge to the eigenvalues of the biharmonic operator on \(\partial\Omega\); in fact, they converge to the eigenvalues of a system of differential equations on \(\partial\Omega\). Based on joint works with J.M. Arrieta, P.D. Lamberti and L. Provenzano.
February 14	Maxime Fortier Bourque (Université de Montréal)
	The multiplicity of \(\lambda_1\) in genus 3
	View Abstract The maximum multiplicity of the first positive eigenvalue \(\lambda_1\) of the Laplacian on Riemannian surfaces has been studied by several authors. Its value is known for surfaces of small complexity, but its asymptotic rate of growth in terms of the complexity is unknown. In this talk, I will discuss joint work with Bram Petri in which we prove that the multiplicity of \(\lambda_1\) is at most 8 for closed hyperbolic surfaces of genus 3 and that this bound is achieved by the most symmetric such surface, the Klein quartic. The proof of the upper bound splits into two parts depending on the value of \(\lambda_1\). When \(\lambda_1\) is not too large, we combine ideas of Sévennec for bounding the multiplicity on arbitrary surfaces with the Faber-Krahn inequality. When \(\lambda_1\) is relatively large, we prove bounds using the Selberg trace formula and rigorous numerical calculations, a strategy adapted from the theory of sphere packings with many potential future applications. As for the lower bound on the multiplicity at the Klein quartic, it uses both trace formula techniques and representation theory.
February 21	Malo Jézéquel (MIT)
	Semiclassical measures for higher dimensional quantum cat maps
	View Abstract Quantum chaos denotes the study of quantum systems whose associated classical dynamics is chaotic. For instance, a central question is to study the high frequencies behavior of the eigenstates of the Laplace-Beltrami operator on a negatively curved compact Riemannian manifold \(M\). In that case, the associated classical dynamics is the geodesic flow on the unit tangent bundle of \(M\), which is hyperbolic and hence chaotic. Quantum cat maps are a popular toy model for this problem, in which the geodesic flow is replaced by a cat map, i.e. the action on the torus of a matrix with integer coefficients. In this talk, I will introduce quantum cat maps, and then discuss a result of delocalization for the associated eigenstates. This result is deduced from a fractal uncertainty principle. Similar statements have been obtained in the context of negatively curved surfaces by Dyatlov-Jin and Dyatlov-Jin-Nonnenmacher, and the case of two-dimensional cat maps have been dealt with by Schwartz. The novelty of our result is that we are sometimes able to bypass the restriction to low dimensions. This is a joint work with Semyon Dyatlov.
February 28	Ilaria Lucardesi (Institut Élie Cartan)
	On the maximization of the first (non trivial) Neumann eigenvalue of the Laplacian under perimeter constraint
	View Abstract In this talk I will present some recent results obtained in collaboration with A. Henrot and A. Lemenant (both in Nancy, France), on the maximization of the first (non trivial) Neumann eigenvalue, under perimeter constraint, in dimension 2. Without any further assumption, the problem is trivial, since the supremum is \(+\infty\). On the other hand, restricting to the class of convex domains, the problem becomes interesting: the maximum exists, but neither its value nor the optimal shapes are known. In 2009 R.S. Laugesen and B.A. Siudeja conjectured that the maximum among convex sets should be attained at squares and equilateral triangles. We prove that the conjecture is true for convex planar domains having two axes of symmetry.
March 7	Nelia Charalambous (University of Cyprus)
	The form spectrum of open manifolds
	View Abstract In this talk we will consider the essential spectrum of the Laplacian on differential forms over noncompact manifolds. We will see a brief overview of known results and discuss the main differences between the function and form spectrum. One interesting problem in the area is finding sufficient and general enough conditions on the manifold so that the essential spectrum on forms is a connected set. We will see that over asymptotically flat manifolds this is the case. The proof involves the study of the structure of the manifold at infinity via Cheeger-Fukaya-Gromov theory and Cheeger-Colding theory, combined with a generalized Weyl criterion for the computation of the spectrum. Finally, we present some recent results on the form spectrum of negatively curved manifolds.
March 14	James Bonifacio (University of Cambridge)
	Bootstrap bounds on closed hyperbolic surfaces
	View Abstract In this talk I will discuss an approach for finding numerical bounds on the eigenvalues of the Laplace-Beltrami operator on closed orientable hyperbolic surfaces that is inspired by a method from physics called the conformal bootstrap. An example of such a bound is that the first nonzero eigenvalue, \(\lambda_1\), must be less than 3.8388977. This bound is almost saturated by the Bolza surface, which is a genus-2 surface with \(\lambda_1 \approx 3.8388873\). Similarly, this approach gives the bound \(\lambda_1 \leq 2.678483\) for closed hyperbolic surfaces of genera at least 3, while the Klein quartic is a genus-3 surface with \(\lambda_1 \approx 2.678\). These bounds were derived using numerical optimisation methods without rigorous error estimates, but they can also be made rigorous. I will discuss how to obtain these bounds, as well as some other applications of this approach.
March 21	Mini Conference: Young researchers in spectral geometry IV
9:30AM PDT 12:30AM EDT 16:30PM BST 17:30 PM CEST	Stine Marie Berge (Leibniz Universität Hannover)
	Convexity Properties for Harmonic Functions on Riemannian Manifolds
	View Abstract In the 70's Almgren noticed that for a harmonic real-valued function defined on a ball, its \(L^2\)-norm over a sub-sphere will have an increasing logarithmic derivative with respect to the radius of mentioned sphere. We examined similar integrals over a more general class of parameterized surfaces by studying harmonic functions defined on compact subdomains of Riemannian manifolds. The integrals over spheres are also generalized to level sets of a given function satisfying certain conditions. If we consider the \(L^2\) norms over these level sets parametrized by a generalization of the radius, we again reproduce Almgren's convexity property. We will sketch the proof of this result and illustrate the usefulness of the convexity result by examining some explicit parameterized families of surfaces, e.g. geodesic spheres and ellipses.
10:15AM PDT 13:15PM EDT 17:15PM BST 18:15 PM CEST	Antoine Métras (Université de Montréal)
	Steklov conformally extremal metrics in higher dimensions
	View Abstract Steklov extremal metrics on surfaces have been much studied due to their connection to free-boundary minimal surfaces found by Fraser and Schoen. In this talk, I will present a characterization of higher dimensional Steklov conformally extremal metrics, highlighting its similarities with the same problem for Laplace eigenvalues. To this end, I will answer the question of which normalization to use and show how the Steklov problem with boundary density appears natural in this context. This is joint work with Mikhail Karpukhin.
11:00AM PDT 14:00PM EDT 18:00PM BST 19:00 PM CEST	Theo McKenzie (UC Berkeley)
	Many nodal domains in random regular graphs
	View Abstract If we partition a graph according to the positive and negative components of an eigenvector of the adjacency matrix, the resulting connected subcomponents are called nodal domains. Examining the structure of nodal domains has been used for more than 150 years to deduce properties of eigenfunctions. Dekel, Lee, and Linial observed that according to simulations, most eigenvectors of the adjacency matrix of random regular graphs have many nodal domains, unlike dense Erdős-Rényi graphs. In this talk, we show that for the most negative eigenvalues of the adjacency matrix of a random regular graph, there is an almost linear number of nodal domains. Joint work with Shirshendu Ganguly, Sidhanth Mohanty, and Nikhil Srivastava.
March 28	Mikhail Cherdantsev (Cardiff University)
	Homogenisation, Spectral Theory, High-contrast Random Composites
	View Abstract We study the homogenisation problem for elliptic operators of the form \(\mathcal A_\varepsilon = -\nabla A^\varepsilon \nabla\) with high-contrast random coefficients \(A^\varepsilon\). In particular, we are interested in the behaviour of their spectra. We assume that on one of the components of the composite the coefficients \(A^\varepsilon\) are "of order one", the complimentary ``soft" component consists of randomly distributed inclusions, whose size and spacing are of order \(\varepsilon \ll 1\), and the values of \(A^\varepsilon\) on the inclusions are of order \(\varepsilon^2\). Our interest in high-contrast homogenisation problems is motivated by the band-gap structure of their spectra. From an intuitive point of view this phenomenon can be explained by looking at the ``soft'' inclusions as micro-resonators, which may dramatically amplify of completely block the propagation of waves in the medium, depending on the frequency. From a mathematically rigorous perspective, this was first analysed by Zhikov (2000, 2004) in the periodic setting. Despite a vigorous activity in the field of periodic high-contrast homogenisation during the last two decades, the stochastic (random) high-contrast setting was largely overlooked, perhaps due to the technical challenges and and more complicated intuitive picture. In this talk I will present our recent results for the stochastic high-contrast setting. First we will look at the homogenised operator \(\mathcal{A}^{\rm hom}\) and describe its spectrum. Then I will give our main results on the convergence of the spectra of \(\mathcal A_\varepsilon\) and a characterisation of the limit spectrum \(\lim_{\varepsilon\to 0}{\rm Sp}(\mathcal{A}^{\rm hom} )\). In contrast with the periodic setting, in the stochastic case the spectrum of the homogenised operator \({\rm Sp}(\mathcal{A}^{\rm hom} )\) is, in general, a proper subset of \(\lim_{\varepsilon\to 0}{\rm Sp}(\mathcal{A}^{\rm hom} )\). We analyse the ``additional'' part of the spectrum - the difference between \(\lim_{\varepsilon\to 0}{\rm Sp}(\mathcal{A}^{\rm hom} )\) and \({\rm Sp}(\mathcal{A}^{\rm hom} )\), and provide its {\it asymptotic} characterisation. Finally, I will touch upon the localisation of defect modes in the gaps of the limiting spectrum \(\lim_{\varepsilon\to 0}{\rm Sp}(\mathcal{A}^{\rm hom} )\) and our current and future work.
April 4	Bruno Colbois (Université de Neuchâtel)
	Upper bounds for Steklov eigenvalues
	View Abstract I will explain two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first is in terms of the extrinsic diameters of the boundary, its injectivity radius and the volume of the manifold. By applying these bounds to cylinders over closed manifold, we obtain new bounds for eigenvalues of the Laplace operator on closed manifolds, in the spirit of Berger—Croke. The second involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its distortion. I will take time to give examples in order to explain why the quantities appearing in the inequalities are necessary. This is a joint work with Alexandre Girouard.
April 11	Jean Lagacé (King's College London)
	Variations on the Weyl law for the Steklov problem on surfaces
	View Abstract Weyl's law describes the distribution of eigenvalues as the spectral parameter goes to infinity. For surfaces with smooth boundary, a superpolynomially precise Weyl law for the Steklov problem for the Laplacian can be obtained via the theory of pseudodifferential operators on circles. Adapting those methods, we obtain a complete asymptotic expansion (as well as spectral invariants) for the eigenvalues of the Steklov problem associated with a Schrödinger operator, and a remainder estimate depending on the regularity when the boundary is not smooth but merely of class \(C^r\).* When the boundary is merely Lipschitz, it is a well-known open problem to prove that the eigenvalues still obey a Weyl law. Following the works of Suslina, and later Agranovich, we know that piecewise \(C^1\) is sufficient to obtain such asymptotics. Using conformal techniques and through a perturbation of the pseudodifferential results we can obtain a Weyl law for the Steklov problem not only for domains with Lipschitz boundary, but also in more singular settings that include every polynomial cusp that is slow enough to obtain a discrete Steklov spectrum.*** In this talk, I will describe the Weyl laws that we have obtained over the last few years, briefly mention the ideas behind the proofs and, time permitting, talk about the directions we could go from there. * Joint work with Simon St-Amant (Cambridge) Joint work with Broderick Causley (McGill) * Joint work with Mikhail Karpukhin (Caltech) and Iosif Polterovich (Montréal)
April 25-29	GEMSTONE mini-course
	Mikhail Karpukhin (California Intitute of Technology) and Daniel Stern (University of Chicago)
	Harmonic maps, minimal surfaces, and shape optimization in spectral geometry
May 2	Pedro Freitas (University of Lisbon)
	Pólya-type inequalities on spheres and hemispheres
	View Abstract We consider the spectra of (round) spheres and hemispheres with the aim of characterising which eigenvalues satisfy Pólya's conjecture and which do not. We then determine correction terms to the first term in the Weyl asymptotics allowing us to provide sharp upper and lower bounds for the corresponding eigenvalues.
May 9	Beniamin Bogosel (Ecole Polytechnique)
	On the polygonal Faber-Krahn inequality
	View Abstract It has been conjectured by Pólya and Szegö in 1951 that among n-gons with fixed area the regular one minimizes the first eigenvalue of the Dirichlet-Laplace operator. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this work we show that the proof of the conjecture can be reduced to finitely many certified numerical computations. Moreover, the local minimality of the regular polygon is reduced to a single validated numerical computation. The steps of the proof strategy include the analytic computation of the Hessian matrix of the first eigenvalue, the stability of the Hessian with respect to vertex perturbations and analytic upper bounds for the diameter of an optimal set. Explicit a priori error estimates are given for the finite element computation of the eigenvalues of the Hessian matrix of the first eigenvalue associated to the regular polygon. Results presented are obtained in collaboration with Dorin Bucur.
May 16	Yves Colin de Verdière (Institut Fourier)
	Attractors for internal or inertial waves
	View Abstract Motivated by experiments of physicists, in particular Thierry Dauxois (ENS-Lyon), we started this work with Laure Saint-Raymond. Using a simplified linear model, we are able to give a precise mathematical status to these attractors. Using Mourre theory and some pseudo-differential calculus, we can show that such attractors exist generically in 2D. [CdV] Spectral theory of pseudodifferential operators of degree 0 and an application to forced linear waves. Anal. PDE 13, No. 5, 1521-1537 (2020). [CdV & Laure Saint-Raymond] Attractors for two-dimensional waves with homogeneous Hamiltonians of degree 0. Commun. Pure Appl. Math. 73, No. 2, 421-462 (2020).
May 23	Dorin Bucur (Université de Savoie)
	Maximization of Neumann Eigenvalues
	View Abstract We discuss the maximization of the \(k\)-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of \({\mathbb R}^N\) with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in \({\mathbb R}^N\) with prescribed mass and prove the existence of an optimal density. For \(k=1,2\) the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For \(k \ge 3\) this question remains open, except in one dimension of the space where we prove that the maximal densities correspond to a union of \(k\) equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the Pólya conjecture in the class of densities in \(\mathbb R\). Based on the relaxed formulation, we provide numerical approximations of optimal densities for \(k=1, \dots, 8\) in \({\mathbb R}^2\). This is a joint work with E. Martinet and E. Oudet.
May 30	Romain Petrides (Institut de mathématiques de Jussieu)
	Minimizing combinations of Laplace eigenvalues and applications
	View Abstract We give a variational method for existence and regularity of metrics which minimize combinations of eigenvalues of the Laplacian among metrics of unit area on a surface. We show that there are minimal immersions into ellipsoids parametrized by eigenvalues, such that the coordinate functions are eigenfunctions with respect to the minimal metrics. As one of the applications, we explain a new method to construct non-planar minimal spheres into 3d-ellipsoids after Haslhofer-Ketover and Bettiol-Piccione.
June 6	Jiaoyang Huang (Courant Institute)
	Extreme eigenvalues of random \(d\)-regular graphs
	View Abstract Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. In particular, the spectral gap, the gap between the first and second largest eigenvalues, measures the expanding property of the graph. In this talk, I will focus on random \(d\)-regular graphs. I'll first explain some conjectures on the extremal eigenvalue distributions of adjacency matrices of random \(d\)-regular graphs; some have been solved, some are still widely open. In the second part of the talk, I will give a new proof of Alon's second eigenvalue conjecture that with high probability, the second eigenvalue of a random \(d\)-regular graph is bounded by \(2\sqrt{d-1}+o(1)\), where we can show that the error term is polynomially small in the size of the graph. This is based on a joint work with Horng-Tzer Yau.
June 13	Dennis Kriventsov (Rutgers University)
	Stability for Faber-Krahn inequalities and the ACF formula
	View Abstract The Faber-Krahn inequality states that the first Dirichlet eigenvalue of the Laplacian on a domain is greater than or equal to that of a ball of the same volume (and if equality holds, then the domain is a translate of a ball). Similar inequalities are available on other manifolds where balls minimize perimeter over sets of a given volume. I will present a new sharp stability theorem for such inequalities: if the eigenvalue of a set is close to a ball, then the first eigenfunction of that set must be close to the first eigenfunction of a ball, with the closeness quantified in an optimal way. I will also explain an application of this to the behavior of the Alt-Caffarelli-Friedman monotonicity formula, which has implications for free boundary problems with multiple phases. This is based on joint work with Mark Allen and Robin Neumayer.
June 20	Andrea Mondino (University of Oxford)
	Optimal transport and quantitative geometric inequalities
	View Abstract The goal of the talk is to discuss a quantitative version of the Levy--Gromov isoperimetric inequality (joint with Cavalletti and Maggi) as well as a quantitative form of Obata's rigidity theorem (joint with Cavalletti and Semola). Given a closed Riemannian manifold with strictly positive Ricci tensor, one estimates the measure of the symmetric difference of a set with a metric ball with the deficit in the Levy--Gromov inequality. The results are obtained via a quantitative analysis based on the localisation method via \(L^1\)-optimal transport. For simplicity of presentation, the talk will present the results in case of smooth Riemannian manifolds with Ricci Curvature bounded below; moreover it will not require previous knowledge of optimal transport theory.
June 27	Nunzia Gavitone (Università degli Studi di Napoli Federico II)
	An eigenvalue problem for the Laplace operator in doubly connected domains
	View Abstract In this talk I will discuss about an eigenvalue problem for the Laplace operator in annular domains \(\Omega = \Omega_0 \setminus \overline{B_{R_1}}\), where \(\Omega_0 \subset \mathbb{R}^n\) is a convex set and \(B_{R_1}\) is the ball of \(\mathbb{R}^n\) centered at the origin with radius \(R_1 > 0\) such that \(\overline{B_{R_1}} \subset \Omega_0\). More precisely Dirichlet and Steklov boundary conditions are imposed on \(\partial B_{R_1}\) and on \(\partial \Omega_0\), respectively. The aim of the talk is to describe the main properties of the first eigenvalue of this problem and to discuss about some related optimization problems. The results I will describe are contained in two joint works with Gloria Paoli, Gianpaolo Piscitelli and Rossano Sannipoli.
July 4	Nadine Große (Universität Freiburg)
	Boundary value problems on domain with cusps
	View Abstract We consider boundary value problems of the Laplacian with Dirichlet (or mixed) boundary conditions on domains with singularities. In two dimensions these singularities include also cusps. Our approach is by blowing up the singularities via a conformal change to translate the boundary problem to one on a noncompact manifold with boundary that is of bounded geometry and of finite width. This gives a natural geometric interpretation in the appearing weights and additional conditions needed to obtain well-posedness results. This is joint work with Bernd Ammann (Regensburg) and Victor Nistor (Universite de Lorraine).
July 11	Melanie Rupflin (University of Oxford)
	Almost harmonic maps from higher genus surfaces
	View Abstract For geometric variational problems one often only has weak, rather than strong, compactness results and hence has to deal with the problem that sequences of (almost) critical points \(u_j\) can converge to a limiting object with different topology. A major challenge posed by such singular behaviour is that the seminal results of Simon on Lojasiewicz inequalities, which are one of the most powerful tools in the analysis of the energy spectrum of analytic energies and the corresponding gradient flows, are not applicable. In this talk we present a method that allows us to prove Lojasiewicz inequalities in the singular setting of almost harmonic maps that converge to a simple bubble tree and explain how these results allow us to draw new conclusions about the energy spectrum of harmonic maps and the convergence of harmonic map flow for low energy maps from surfaces of positive genus into general analytic manifolds.