Spectral Geometry in the Clouds
Intergalactic seminar
on spectral theory and geometric analysis
on spectral theory and geometric analysis
| Home | Program | Useful Links |
| September 23 | Marcello Seri (University of Groningen) |
| A panorama of singular sub-Laplacians and their spectra | |
View AbstractLaplace-Beltrami operators on rank-varying sub-Riemannian structures have been recently gaining interest due to their exotic properties; this talk is an invitation for the audience to explore them. We will start from the 0th property of their analysis: self-adjointness. In many cases, and in contrast with the Riemannian case, the sub-Riemannian setting presents large families of operators which are essentially self-adjoint even though the manifold is non-complete. We will then move on to present a brief panoramic view of what is known about their spectral properties, with a particular emphasis on sub-Riemannian Weyl laws. Throughout the talk we will touch upon a number of simple-to-state open questions to stimulate interest and foster future discussions. |
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| September 30 | Lyonell Boulton (Heriot Watt University) |
| Spectral analysis of Dirac operators with a purely imaginary dislocation | |
View AbstractIn this talk we present a complete spectral analysis of Dirac operators in 1D with non-Hermitian matrix potentials of the form \(i\mathrm{sgn}(x) +V (x)\) where \(V \in L^1\). For \(V = 0\) we compute explicitly the matrix Green function. This allows us to determine the spectrum, which is purely essential, and its different types. It also allows us to find sharp enclosures for the pseudospectrum and its complement, in all parts of the complex plane. Notably, this includes the instability region, corresponding to the interior of a band surrounding the real axis. Then, with the help of a Birman-Schwinger principle, we establish in precise manner how the spectrum and pseudospectrum change when \(V\ne 0\), assuming the hypotheses \(\|V\|_{L^1} <1 \) or \(V \in L^1 \cap L^p \) where \(p > 1\). We show that the essential spectra remain unchanged and that the \(\varepsilon\)-pseudospectrum stays close to the instability region for small \(\varepsilon\). We determine sharp asymptotic estimates for the discrete spectrum, whenever \(V\) satisfies further conditions of decay at infinity. Finally, in one of our main findings, we give a complete description of the weakly-coupled model. The research leading to these results was conducted in collaboration with Tho Nguyen Duc and David Krejčiřík, from the Czech Technical University in Prague. |
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| October 7 | Laura Monk (University of Bristol) |
| The moduli space of twisted Laplacians and random matrix theory | |
View AbstractRudnick recently proved that the spectral number variance for the Laplacian of a large compact hyperbolic surface converges, in a certain scaling limit and when averaged with respect to the Weil-Petersson measure on moduli space, to the number variance of the Gaussian Orthogonal Ensemble of random matrix theory. In this talk, I will present joint work with Jens Marklof, where we extend Rudnick’s approach to show convergence to the Gaussian Unitary Ensemble for twisted Laplacians which break time-reversal symmetry, and to the Gaussian Symplectic Ensemble for Dirac operators. This addresses a question of Naud, who obtained analogous results for twisted Laplacians on high genus random covers of a fixed compact surface. |
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| October 14 | Amir Vig (University of Michigan) |
| Marked length spectral invariants of Birkhoff billiard tables and compactness of isospectral sets | |
View AbstractFor planar billiard tables, the marked length spectrum encodes the lengths of action (minus the length) minimizing orbits of a given rational rotation number. For strictly convex tables, a renormalization of these lengths extends to a continuous function called Mather's beta function (or the mean minimal action). We show that using the algebraic structure of its Taylor coefficients, one can prove \(C^\infty\) compactness of marked length isospectral sets. This gives a dynamical counterpart to the Laplace spectral results of Melrose, Osgood, Phillips and Sarnak. |
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| October 21 | Julie Rowlett (Chalmers University) |
| Spectral invariants of integrable polygons | |
View AbstractAn integrable polygon is one whose interior angles are of the form \(\pi/n\) for positive integers \(n\). As an exercise, can you determine all integrable polygons? I’ll discuss joint work with Gustav M˚ardby in which we obtain new spectral invariants for these polygons. It turns out that integrable polygons are also precisely those polygons which strictly tessellate the plane. I will also discuss joint work with M. Blom, H. Nordell, O. Thim, and J. Vahnberg in which we explore the generalization of integrable polygons to higher dimensions. This includes an equivalence between the geometric characterization of strict tessellation, an analytic characterization of Dirichlet Laplace eigenfunctions, and an algebraic characterization of crystallographic Coxeter groups. |
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| October 28 | Irving Calderón (Durham University) |
| Spectral gap for random Schottky surfaces | |
View AbstractFor decades, the study of the spectrum of the Laplacian of Riemannian manifolds has been a very active topic of research at the crossroads of Geometry, Dynamics, Number Theory and Probability. The particularly rich and beautiful theory for hyperbolic surfaces (i.e. with constant curvature \(-1\)) holds a privileged spot in the area because it deals with objects that are explicit enough to allow us to get our hands-on, yet it still holds many mysteries. One of the broad goals of the area is to understand the behaviour of the Laplace eigenvalues of a "typical" hyperbolic surface. In this talk I will present a spectral gap result for random hyperbolic surfaces of infinite area without cusps (aka Schottky surfaces), obtained in collaboration with M. Magee and F. Naud. Our result can be interpreted as a probabilistic analog for Schottky surfaces of Selberg’s celebrated \(1/4\)-Conjecture. |
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| November 4 | Matthias Hofmann (FernUniversität in Hagen) |
| Graph structure of the nodal set on Riemannian manifolds | |
View AbstractWe illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory, vanishing order of eigenfunctions, and unique continuation. We identify the nodal set of an eigenfunction of the Schrödinger operator (with smooth potential) on a compact, orientable Riemannian manifold as an imbedded metric graph and then use tools from elementary graph theory in order to estimate the number of critical points in the nodal set of the \(k\)-th eigenfunction and the sum of vanishing orders at critical points in terms of \(k\) and the genus of the manifold. Based on a joint work with Matthias Täufer. |
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| November 8 - 13 | GEMSTONE mini-course |
| Nilima Nigam (Simon Fraser University) | |
| Approximation Strategies for spectral problems | |
| November 18 | Martijn Kluitenberg (University of Groningen) |
| Cheeger's inequality in sub-Riemannian geometry | |
View AbstractWe explore extending the classical Cheeger inequality to sub-Riemannian manifolds, which are manifolds in which shortest paths can only take velocities confined to a subbundle of the tangent bundle. During the talk, I will discuss the basis properties of sub-Riemannian structures, as well as the key geometric and analytic concepts used in the proof of Cheeger’s inequality. Based on arXiv:2312.13058. |
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| November 25 | Charlotte Dietze (Ludwig-Maximilians Universität München) |
| Weyl formulae for some singular metrics with application to acoustic modes in gas giants | |
View AbstractWe prove eigenvalue asymptotics of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of soundwaves in gas planets. This is joint work with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat. |
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| December 2 | Jeff Galkowski (University College London) |
| The Steklov problem with piecewise smooth boundary | |
View AbstractThe Steklov problem is well-studied on domains with smooth boundary, where its eigenfunctions and eigenvalues behave similarly to Laplace eigenfunctions and eigenvalues. However, much less is known when the boundary is singular. In this talk, we will discuss an approach based on scattering theory for the study of the Steklov problem on domains with piecewise smooth, Lipschitz boundaries. We will apply this method to study the eigenvalues and eigenfunctions of the Steklov problem on piecewise smooth, Lipschitz domains in dimension two, where we are able to improve on earlier work of Levitin–Parnovski–Polterovich–Sher. We then indicate some of the further developments needed to understand the problem in higher dimensions. Based on joint work with M. Malagutti and R. Wang. |
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| December 9 | Hanna Kim (University of North Carolina at Chapel Hill) |
| Upper bounds on the second Laplacian eigenvalue on the projective space | |
View AbstractWe investigate the isoperimetric inequality of the second eigenvalue on real projective space in all dimensions. It was shown in two dimensions by Nadirashvili and Penskoi that the second eigenvalue attains a maximum when a sequence eigenvalues approaches to that of a disjoint union of a sphere and a real projective space with a certain ratio. The conjecture is that the result in two dimensions can be generalized to all dimensions within the conformal class of the round metric. We prove an upper bound when the metric degenerates to that of two projective spaces. We construct trial functions for variational characterization by using a generalized Veronese map to a higher dimensional spheres. We also discuss about a topological degree method and its another applications (joint work with R. Laugesen). |
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| December 16 | Yannick Bonthonneau (Université Paris-XIII) |
| 2D pure magnetic tunneling formula | |
View AbstractI will present results in collaboration with Fournais, Morin, Raymond. Concerning the 2D magnetic laplacian, in the presence of non degenerate wells, and a symmetry (and some further assumptions), we obtain an equivalent for the exponential splitting between the first two eigenvalues. |
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| January 13 | Rupert Frank (Ludwig-Maximilians Universität München) |
| Asymptotic optimization of Riesz means of Laplace eigenvalues on convex sets | |
View AbstractWe consider the problem of optimizing Riesz means \(\mathrm{Tr}(−\Delta − \Lambda^\gamma_-)\) of Laplace eigenvalues among convex sets in \(\mathbb{R}^d\) with given measure. More precisely, we maximize Riesz means of Dirichlet eigenvalues and minimize Riesz means of Neumann eigenvalues. We are interested in the behavior of optimizers in the asymptotic regime where \(\Lambda \to \infty\). In 2D we prove convergence in Hausdorff distance to a disk for any Riesz exponent \(\gamma > 0\). We have similar results, either conditional or unconditional, in arbitrary dimension \(d \ge 3\). Our proofs combine uniform versions of Weyl asymptotics with the partially semiclassical analysis of degenerating convex sets. The talk is based on joint work with Simon Larson. |
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| January 20 | Will Hide (University of Oxford) |
| Small eigenvalues of many cusped hyperbolic surfaces | |
View AbstractBased on joint work with Joe Thomas (Durham) We study the spectrum of the Laplacian for non-compact finite-area hyperbolic surfaces, focusing on eigenvalues below \(\frac 14\) which are known as small eigenvalues. A result of Otal and Rosas says that any hyperbolic surface of genus \(g\) with \(n\) cusps has at most \(2g + n − 2\) small eigenvalues, and this bound is sharp. On the other hand, by a theorem of Zograf, if the number of cusps is much larger than the genus then there is necessarily at least \(1\) non-zero small eigenvalue. This provides a topological lower bound on the number of small eigenvalues for many cusped surfaces. I will discuss joint work with Joe Thomas where we show that when \(n\) is much larger than \(g\), any hyperbolic surface of signature \((g, n)\) has at least \(\propto\frac{2g+n}{\log(2g+n)}\) small eigenvalues. |
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| January 27 | Young researchers in spectral geometry IV |
| Tirumala Chakradhar (University of Bristol) | |
| Steklov eigenvalue bounds for differential forms | |
View AbstractWe consider a generalisation of the Steklov problem to the framework of differential forms and present geometric eigenvalue bounds for hypersurfaces of revolution and other warped product manifolds, comparing with the analogous results for functions (i.e., 0-forms). |
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| Maria Esteban Casadevall (University of Oxford) | |
| Reverse isoperimetric properties of thick \(\lambda\)-sausages in the Hyperbolic plane and Blaschke's Rolling Theorem | |
View AbstractIn this talk, we will explore the reverse isoperimetric inequality for convex bodies with uniform curvature constraints in the hyperbolic plane \(\mathbb{H}^2\) . This problem is concerned with finding the body that minimizes the volume for a given surface area. We will begin by stating the research question and highlighting the unique challenges posed by hyperbolic geometry, which makes this problem more challenging than in the Euclidean setting. Next, we will outline the two main theorems of the project and provide an overview of the key ideas used in their proofs. |
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| Samuel Audet-Beaumont (Université Laval) | |
| Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity | |
View AbstractThe Steklov problem offers a wide range of unanswered questions. Amongst them, the study of the multiplicity of the first Steklov eigenvalue remains to be understood. The Steklov case shares some similarities with the classical Laplacian case, one of them being the boundedness of the multiplicity in terms of genus in dimension 2. The general optimal bound still remains to be found in both cases, although some conjectures do exist. The main tools used to prove these results for the Steklov problem are adaptations of the same tools used for the Laplacian, albeit with added complexities. Similarly, we show how to construct a sequence of surfaces with growing genus and first Steklov eigenvalue multiplicity using adaptations of techniques previously used by B. Colbois and P. Buser for the Laplacian. These new constructions show the non-existence of an universal bound, independent of genus, and provide evidence on the possible asymptotical growth of the optimal bound on the multiplicity of the first Steklov eigenvalue. |
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| Chia-Chun Lo (University of Oxford) | |
| Homogenisation for the Robin eigenvalue problem on manifolds and flexibility of optimal Schrödinger potentials | |
View AbstractWe show that the spectrum of a Schrödinger eigenvalue problem on a Riemannian manifold \(M\) can be approached by that of a family of Robin eigenvalue problems posed on a sequence of domains in \(M\). These domains take the form of \(M\) with many small balls removed from it: this is an example of a procedure known as homogenisation, which has recently found numerous applications in the investigation of spectral problems. As an application of our result, we prove a flexibility result for optimal Schrödinger potentials. |
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| Loïs Delande (Université de Bordeaux) | |
| Sharp spectral gap for degenerated Witten Laplacian | |
View AbstractIn this talk I will state a precise description of the small eigenvalues of the Witten Laplacian associated to a non-Morse potential. Relying on the construction of sharp degenerated Gaussian quasimodes and an adaptation of the WKB method, I can prove Eyring-Kramers formulae for the bottom of the spectrum and quatify the spectral gap in the semiclassical regime between these eignevalues and the rest of the spectrum. |
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| February 3 | Charles Bordenave (Aix-Marseille Université) |
| Random perturbation of Toeplitz matrices | |
View AbstractToeplitz matrices form a rich and ubiquitous class of possibly non-normal matrices. Their asymptotic spectral analysis in high dimension is wellunderstood, as illustrated by the strong Szegő limit theorem for Toeplitz determinants. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this talk, we explore the spectrum of a banded Toeplitz matrix perturbed by a random matrix in the asymptotic of high dimension. We show that the outlier eigenvalues are driven by a lowdimensional random analytic matrix field alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. Along the way, we present new variations around the strong Szegő limit theorem. The talk is based on a joint work with Mireille Capitaine and François Chapon. |
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| February 10 | Luigi Provenzano (Sapienza Università di Roma) |
| Isoperimetric and geometric inequalities for the magnetic Laplacian | |
View AbstractWe consider the eigenvalues of the magnetic Laplacian on bounded domains of \(\mathbb{R}^2\) with uniform magnetic field \(\beta > 0\) and magnetic Neumann boundary conditions. We discuss upper and lower bounds for the ground state energy \(\lambda_1\) and a reverse Faber-Krahn inequality for simply connected domains. Based on joint works with Bruno Colbois (Université de Neuchâtel), Corentin Léna (Università degli Studi di Padova) and Alessandro Savo (Sapienza Università di Roma). |
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| February 17 | Julien Moy (Université Paris-Saclay) |
| Spectral statistics of negatively curved surface covers | |
View AbstractIn the early 1980s, Bohigas, Giannoni and Schmit formulated a conjecture (BGS) about the spectral distribution of quantum systems whose classical limit is chaotic. They proposed that generically, in the high energy limit, such systems should display spectral statistics predicted by Random Matrix Theory (RMT). Although some numerical experiments and heuristic arguments based on trace formulae support the BGS conjecture, little to no progress towards a rigorous proof has been made. Some recent developments have focused on random models of quantum systems, for which one may hope to prove results with high probability, e.g. for 99% of systems. In this talk, I will discuss some results on the spectral distribution of the Laplacian on random covers of a surface of variable negative curvature. In the limit of large degree, the (smoothed) counting function of eigenvalues is shown to display fluctuations predicted by RMT. |
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| March 3 | Jean Lagacé (King's College London) |
| Foliations of bubblesheet singularities: a spectral geometric approach | |
View AbstractAt the core of differential geometry is the notion that the important features of a space should remain invariant under changes of coordinates. Neverthe- less, spaces with special structure may admit preferred coordinate systems, highlighting some of its features with particular clarity. Such distinguished parameterisations have often been found by identifying a foliation of the space by submanifolds canonically determined by its geometry. An exam- ple is foliations by constant mean curvature (CMC) hypersurfaces, which have been used for instance to parameterise the ends of asymptotically flat manifolds, leading to a definition of center of mass for isolated gravitating systems. They also played a crucial role in the first proof of the stability of Minkowski spacetime, or in foliating geometric ”necks” to continue geometric flows through neck singularities via surgery. In the codimension \(n \ge 2\) setting, the situation is more complicated. In- deed, where the CMC condition would naturally be replaced by Parallel Mean Curvature (PMC), there are generic geometric obstructions for the establish- ment of such a foliation. In this work, we introduce a new, pseudodiffer- ential, curvature condition, which we dub ”Quasi-Parallel Mean Curvature” (QPMC), and find that bubblesheet singularities (the higher codimension counterpart to necks) can be foliated by QPMC embedded spheres. I will present this curvature condition built on eigenspaces of the connection Lapla- cian and the construction of the foliation, as well as examples that indicate the necessity of such a condition. Time permitting, I may present some applications to Mean Curvature Flow. |
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| March 10 | Theo Mckenzie (Stanford University) |
| Precise eigenvalue location for random regular graphs | |
View AbstractThe spectral theory of regular graphs has broad applications in theoret- ical computer science, statistical physics, and other areas of mathematics. Graphs with optimally large spectral gap are known as Ramanujan graphs. Previous constructions of Ramanujan graphs are based on number theory and have specific constraints on the degree and number of vertices. In this talk, we show that, in fact, most regular graphs are Ramanujan; specifically, a randomly selected regular graph has a probability of 69% of being Ramanu- jan. We establish this through a rigorous analysis of the Green’s function of the adjacency operator, focusing on its behavior under random edge switches. |
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| March 17 | Jaume de Dios Pont (ETH Zürich) |
| Convex sets can have interior hot spots | |
View AbstractA homogeneous, insulated object with a non-uniform initial temperature will eventually reach thermal equilibrium. The Hot Spots conjecture addresses which point in the object takes the longest to reach this equilibrium: Where is the maximum temperature attained as time progresses? Rauch initially conjectured that points attaining the maximum temperature would approach the boundary for larger times. Burdzy and Werner disproved the conjecture for planar domains with holes. Kawohl, and later Bañuelos–Burdzy, conjectured that the conjecture should still hold for convex sets of all dimensions. This talk will draw inspiration from a recurrent theme in convex analysis: almost every dimension-free result in convex analysis has a natural log-concave extension. We will motivate and construct the log-concave analog of the Hot Spots conjecture, and then disprove it. Using this log-concave construction, we will show that the hot spots conjecture for convex sets is false in high dimensions. |
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| March 24 | Benjamin Florentin (Institut Élie-Cartan de Lorraine) |
| Steklov spectral inverse problem in a conformal class | |
View AbstractIn this talk, we will focus on the spectral inverse problem consisting in recovering the metric of a compact Riemannian manifold with boundary from knowledge of its Steklov spectrum, or equivalently the spectrum of its Dirichlet-to-Neumann map. In other words, can one hear the shape of a “Steklov drum”? When the boundary is Anosov with simple length spectrum, the study of singularities in the trace of the wave operator makes it possible to exhibit certain spectral invariants via the Duistermaat-Guillemin trace formula and we will discuss how these invariants can be exploited and naturally combined with the injectivity of the geodesic X-ray transform to attack the problem. Some results obtained in the class of conformal metrics will be presented and it will be explained how generically it is possible to “hear the Taylor series at the boundary” of a conformal Steklov drum. |
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| March 31 | Phanuel Mariano (Union College) |
| On a conjecture of a Pólya functional for triangles and rectangles | |
View AbstractWe consider a functional involving the product of the first Dirichlet eigen- value \(\lambda_1(\Omega)\) and the torsional rigidity \(T(\Omega)\) of a planar domain \(\Omega\) normalized by its area \(|\Omega|\). This functional was originally considered by Pólya who first showed that this quantity is bounded by 1. It has been conjectured that this functional is bounded above by \(\frac{\pi^2}{12}\) and below by \(\frac{\pi^2}{24}\) over the class of bounded planar convex domains. We prove this is true for the class of all tri- angles and rectangles. In particular, we prove precise estimates for triangles and a monotonicity property for rectangles. This talk is based on joint work with Rodrigo Bañuelos. |
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| April 7 | Isabel Fernández (University of Seville) |
| Free boundary minimal annuli in the ball | |
View AbstractThe theory of free boundary minimal surfaces in the unit ball of R3 is closely related to the Steklov problem. Utilizing this connection, Fraser and Schoen established new properties and introduced new examples of such sur- faces in 2016, sparking renewed interest in the field. Since then, numerous results have been obtained by various researchers. In this talk, we will review some of these recent developments, focusing on free boundary minimal annuli and the Critical Catenoid Conjecture, one of the most significant open problems in the theory. |
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| April 14 | Jenya sapir (Binghamton University) |
| Short curves on expander surfaces | |
View AbstractWe discuss what a ”typical” short curve on a random large genus hyperbolic surface looks like. In particular, for each \(L\), there are finitely many curves of length at most \(L\). We find length scales at which such a curve chosen at random is highly likely to be non-simple, or fill the whole surface. It is known that, with respect to many commonly studied random models, a typical surface will be expander. That is, it will be “highly connected,” in the sense that we get effective mixing of the geodesic flow. We will give results that hold for all expander surfaces, and hence for random surfaces with respect to many different random models. This is joint work with Ben Dozier. |
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| April 28 | Gabriel Rivière (Université de Nantes) |
| Quantum unique ergodicity for magnetic Laplacians on the torus | |
View AbstractThe aim of this talk will be to describe the asymptotic properties of the eigenfunctions of magnetic Laplacians on the flat torus of dimension 2. I will explain that, under a geometric control condition on the magnetic field, the eigenfunctions satisfy equidistribution properties as the eigenvalue tends to infinity. This is a joint work with Léo Morin (Copenhagen). |
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| May 5 | Alba dolores García-Ruiz (ICMAT) |
| Quantum unique ergodicity for magnetic Laplacians on the torus | |
View AbstractIn this talk, we consider a bounded domain in the Euclidean plane and examine the Laplace eigenvalue problem with specific boundary conditions. A famous conjecture by Berry suggests that in chaotic dynamical systems, eigenfunctions resemble random monochromatic waves; however, this behavior is generally not expected in integrable dynamical systems. Here, we explore the behavior of high-energy eigenfunctions and their connection to Berry’s random wave model. In particular, we study a related property, which we call Inverse Localization, describing how eigenfunctions can approximate monochromatic waves in small regions of the domain. |
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| May 9 - 13 | GEMSTONE mini-course |
| Chris Judge (Indiana University, Bloomington) | |
| Variations of Geometry and Spectrum |