Program | Spectral Geometry in the Clouds

Program of the seminar for the Academic Year 2025 to 2026 (first term)

October 6	Lawford Hatcher (Indiana University, Bloomington)
	Comparing Dirichlet and Neumann Laplace spectra
	View Abstract Understanding the relationship between eigenvalues of the Laplacian coming from different boundary conditions is a problem that dates back at least to the work of Polya, who showed that the second Neumann eigenvalue of a bounded planar domain is strictly less than the first Dirichlet eigenvalue. Over the following decades, several extensions of this result to higher dimensions, higher eigenvalues, and to curved spaces have been made by various authors. These inequalities have proved useful in the notoriously difficult problem of understanding qualitative features of eigenfunctions. Inspired by a conjecture of Cox-MacLachlan-Steeves, we explore the relationship between the isoperimetric ratio and the number of Neumann eigenvalues not exceeding the first Dirichlet eigenvalue in three settings: convex bodies of any dimension, polygonal domains, and tubular neighborhoods of Riemannian manifolds. In all three cases, we establish that the isoperimetric ratio of the geometric object gives two-sided bounds on this quantity. We will also discuss counterexamples showing that these estimates cannot be extended to general non-convex domains, implying that the above mentioned conjectures are both false in general.
October 13	Lukas Bundrock (University of Alabama)
	Spectral Optimization in Exterior Domains: Robin and Steklov Problems
	View Abstract The Steklov problem on bounded domains is a classical topic with deep connections to shape optimization, geometric inequalities, and various physical applications, such as modeling diffusion processes where particles exhibit specific behavior upon reaching a boundary. A natural extension is to consider diffusion in the exterior of a domain, leading to the exterior Steklov problem. After introducing suitable formulations for this problem, we discuss several equivalent characterizations and limiting approaches. These tools allow us to establish analogues of classical inequalities, such as Weinstock’s, and to derive an Escobar-type lower bound for the first eigenvalue. This bound reveals that, in contrast to the interior case, isoperimetric inequalities fail in the exterior setting in higher dimensions—even for smooth convex domains. In the second part, we turn to the exterior Robin problem. Building on the results for the Steklov case, we show that similar isoperimetric failures occur for the Robin spectrum in higher dimensions. The results are based on joint work with Alexandre Girouard, Denis Grebenkov, Michael Levitin, and Iosif Polterovich.
October 20	Dorin Bucur (Université de Savoie)
	The geometric size of the fundamental gap
	View Abstract The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in \(\mathbb{R}^N\). The question concerning the rigidity of the inequality, raised by Yau in 1990, was left open. Going beyond rigidity, the result presented in this talk strengthens the Andrews-Clutterbuck inequality, by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The proof relies on a localized, variational interpretation of the fundamental gap, allowing a dimension reduction via the use of convex partitions à la Payne-Weinberger: the result stems by combining a new sharp result for one dimensional Schrödinger eigenvalues with measure potentials, with a thorough analysis of the geometry of the partition into convex cells. As a by-product of our approach, we obtain a quantitative form of Payne-Weinberger inequality for the first nontrivial Neumann eigenvalue of a convex set in \(\mathbb{R}^N\), thus proving, in a stronger version, a conjecture from 2007 by Hang-Wang. This is a joint work with Vincenzo Amato and Ilaria Fragalà.
October 27	Guenda Palmirotta (Universität Paderborn)
	Patterson-Sullivan distributions of hyperbolic surfaces
	View Abstract There is a curious relation between two kinds of phase space distributions associated to eigenfunctions of the Laplacian on a hyperbolic surface: Patterson-Sullivan distributions, which are invariant under the geodesic flow, and Wigner distributions, which arise in quantum chaos and are invariant under the wave group. In this talk, we will describe these two distributions and generalise them on convex-cocompact hyperbolic surfaces. Then, we will show how they are asymptotically intertwined. This is a joint work with Benjamin Delarue (Universität Paderborn). ⚠️ Please note that the seminar will exceptionally take place one hour later in America (9am PDT / 12am EDT). This corresponds to the usual time in Europe (4pm GMT / 5pm CET).
November 3	Denis Vinokurov (Université de Montréal)
	Maximizing Laplace eigenvalues with density in higher dimensions
	View Abstract We will discuss the problem of maximizing the \(k\)-th Laplace eigenvalue with density on a closed Riemannian manifold of dimension \(m \ge 2\). The Euler-Lagrange equation identifies critical densities with the energy densities of harmonic maps into spheres, linking spectral optimization to harmonic-map theory. Unlike the case \(m = 2\), where a priori multiplicity bounds yield existence and regularity, higher dimensions allow unbounded multiplicities. In the talk, we will present techniques from topological tensor products that handle this setting and prove the existence of maximizing densities for all \(m \ge 3\). For regularity, optimizers are smooth away from a singular set; when \(m \ge 7\), this set can have any prescribed integer dimension up to \(m - 7\), as we will illustrate with examples on the \(m\)-sphere. These techniques have potential for other eigenvalue-optimization problems in higher dimensions where unbounded multiplicities arise. References: D. Vinokurov, Maximizing higher eigenvalues in dimensions three and above, preprint, arXiv:2506.09328 [math.SP] (2025).
November 10	Maurizia Rossi (University of Milano-Bicocca)
	The nodal length of random spherical harmonics
	View Abstract In this talk we investigate the behavior of the "typical" Laplacian eigenfunction of a compact smooth Riemannian manifold. In particular, motivated by both Yau's conjecture on nodal sets and Berry's ansatz on planar random waves, we consider Gaussian eigenfunctions on the sphere and study the distribution of the length of their nodal lines in the high energy limit. This result raises several questions regarding both the distribution of other geometric functionals and the behavior of nodal statistics of random eigenfunctions on a "generic" manifold. (This talk is mainly based on a joint work with D. Marinucci and I. Wigman.)
November 17	Jared Wunsch (Northwestern University)
	Gutzwiller trace formula for singular potentials
	View Abstract The Gutzwiller trace formula relates the asymptotic spacing of quantum-mechanical energy levels in the semiclassical limit to the dynamics of periodic classical particle trajectories. We generalize this result to the case of non-smooth potentials, for which there is partial reflection of energy from derivative discontinuities of the potential. It is the periodic trajectories of an associated branching dynamics that contribute to the trace asymptotics in this more general setting; we obtain a precise description of their contribution. This is joint work with Mengxuan Yang and Joey Zou.
November 24	Romain Speciel (Stanford University)
	Commutativity properties of the Dirichlet-to-Neumann map
	View Abstract The Dirichlet-to-Neumann map records how boundary values of harmonic functions determine their normal derivatives, and is a fundamental object of study in inverse problems and spectral geometry. A natural question is: what geometric conditions are forced on the interior by symmetries or spectral features of this boundary operator? In this talk, we will discuss recent results showing that when the Dirichlet-to-Neumann map commutes with the boundary Laplacian, this strongly constrains the underlying geometry, forcing it to be close to the standard ball. We will then explore stability versions of this statement, and discuss applications in inverse problems. The talk is based on two recent papers: [arXiv:2503.00270] and [arXiv:2510.08822].
December 1	Joe Thomas (Durham University)
	Spectral gap for random hyperbolic surfaces
	View Abstract The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about a surface. In this talk, I will discuss the typical size of the spectral gap for a random surface with large genus sampled with respect to the Weil-Petersson probability measure. In particular, I will explain joint work with Will Hide and Davide Macera where we obtain a spectral gap with a polynomial error rate. Our result uses a fusion of the polynomial method used in recent breakthroughs on the strong convergence of group representations with the trace formula for hyperbolic surfaces.
December 8	Iosif Polterovich (Université de Montréal)
	Courant, Bézout, and topological persistence
	View Abstract Functions exhibiting highly oscillatory behaviour naturally appear in different areas of mathematics. In this talk, we discuss extensions of two classical results: Courant's nodal domain theorem for Laplace eigenfunctions and Bézout's theorem for polynomials. Our approach is based on ideas originating in topological persistence. Based on joint work with L. Buhovsky, J. Payette, L. Polterovich, E. Shelukhin, and V. Stojisavljević.
December 15	Daniel Stern (Cornell University)
	Electrostatics and minimal surface doublings
	View Abstract I'll discuss joint work with Adrian Chu, relating Kapouleas's doubling construction for minimal surfaces to the variational theory for a Coulomb-type interaction energy for Schroedinger operators. Namely, for the Jacobi operator of a given nondegenerate minimal surface, we show that families of nondegenerate critical points of this energy give rise to high-genus minimal surfaces approximating the initial surface with multiplicity two, provided a few key estimates are satisfied. By studying the ground states for this interaction energy, we show that, generically, every minimal surface of index one admits such a doubling, and deduce as a corollary that generic 3-manifolds contain sequences of embedded minimal surfaces with bounded area and arbitrarily large genus.
January 19	INI joint seminar
	Eugenia Malinnikova (Stanford University)
	Spectral Inequalities and Quantitative Unique Continuation for Schrödinger operators
	View Abstract Spectral inequalities quantify how strongly a linear combination of low-energy eigenfunctions can concentrate away from a prescribed observation set. In Fourier analysis ,the classical counterpart is the Logvinenko-Sereda theorem, where thickness of the observation set is a natural geometric condition. I will discuss spectral inequalities for confining one-dimensional Schrödinger operators with rough potentials, and some analytic tools behind them. I will also highlight open problems, including sharpness of the geometric hypothesis and extensions to high-dimensional. The talk is based on a joint work with Jiuyi Zhu.
January 26	Mehdi Eddaoudi (Université Laval)
	Payne–Pólya–Weinberger inequalities on closed Riemannian manifolds
	View Abstract The classical Payne–Pólya–Weinberger inequality provides a universal upper bound for the ratio of eigenvalues of the Laplacian on a Euclidean domain, with Dirichlet boundary conditions. On a closed Riemannian manifold, this ratio is generally unbounded, even within a fixed conformal class. In this talk, I will discuss this problem within a broader context of inequalities closely related to Hebey’s A–B program on Sobolev inequalities, and I will explain how this approach naturally leads to the study of eigenvalue ratios for the conformal Laplacian. As an example, I will present a PPW inequality analogous to the celebrated result of El Soufi and Ilias, whose equality case is characterized by minimal immersions into a sphere by first eigenfunctions via the Yamabe metric. Finally, motivated by recent work of Humbert, Petrides, and Premoselli, I will discuss a possible extension of PPW-type inequalities to the full family of GJMS operators and their connections with Q-curvature. This talk is based on joint work in progress with Erwann Aubry and Romain Petrides.
February 9	Roméo Leylekian (Instituto Superior Técnico)
	Payne’s nodal line conjecture fails on doubly-connected planar domains
	View Abstract I will construct a bounded planar domain with one single hole for which the nodal line of a second Dirichlet eigenfunction is closed and does not touch the boundary. This shows that Payne’s nodal line conjecture (1967) can at most hold for simply-connected domains in the plane. I will also mention the case of Neumann boundary conditions.
February 23	INI joint seminar
	Nunzia Gavitone (Università degli Studi di Napoli Federico II)
	Overdetermined problems and higher order mean curvatures: symmetry and stability results.
	View Abstract In this talk I will speak about a deep connection between Serrin’s symmetry result for Hessian operators and constant higher order mean curvature boundaries.The result i will describe are contained in a joint work with A.L. Masiello, G. Paoli and G. Poggesi. Our analysis will not only provide new proofs of the higher order Soap Bubble Theorem but also bring several benefits, including new interesting symmetry results and quantitative stability estimates.
March 2	Mitchell Taylor (ETH Zürich)
	On the exact failure of the hot spots conjecture
	View Abstract The hot spots conjecture asserts that as time goes to infinity, the hottest and coldest points in an insulated domain will migrate towards the boundary of the domain. In this talk, I will describe joint work with Jaume de Dios Pont and Alex Hsu where we find the exact failure of the hot spots conjecture in every dimension.
March 9	Alexander Strohmaier (Leibniz Universität Hannover)
	Interactions of spectral geometry and general relativity
	View Abstract I will discuss some questions that appear naturally in quantum field theory on curved spacetimes but are spectral theory problems in disguise. This includes a discussion as to what spectral invariants are expected to be important in this context.
March 16	INI joint seminar
	Daniel Grieser (Universität Oldenburg)
	The Steklov spectrum for spaces with fibred cusp boundary
	View Abstract The Dirichlet-Neumann operator for a compact Riemannian manifold with smooth boundary has discrete spectrum, known as Steklov spectrum. This still holds if the boundary is only Lipschitz. However, as was first analyzed by Nazarov and Taskinen, essential spectrum appears when the boundary has cusp singularities. I will consider the more general case of fibred cusp singularities (a simple example being the domain obtained by taking a ball B and removing from it a smaller ball touching B from the inside). I will give a precise description of the Schwartz kernel of the Dirichlet-Neumann operator near the singularity -- it lies in an adapted singular pseudodifferential calculus, the phi-calculus of Mazzeo and Melrose -- and a formula for the bottom of its essential spectrum. This is joint work with K. Fritzsch and E. Schrohe. ⚠️ Please note that the seminar will exceptionally take place one hour later in most of America (9am PDT / 12am EDT). This corresponds to the usual time in Europe (4pm GMT / 5pm CET).
April 27	Yunhui Wu (Tsinghua University)
	Uniform spectral gaps for random hyperbolic surfaces with not many cusps
	View Abstract We investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if \(n=O(g^\alpha)\) where \(\alpha\in \left[0,\frac{1}{2}\right)\), then for any \(\epsilon>0\), a random cusped hyperbolic surface in the moduli space of cusped hyperbolic surfaces of genus \(g\) with \(n\) punctures has no eigenvalues in \(\left(0,\frac{1}{4}-\left(\frac{1}{6(1-\alpha)}\right)^2-\epsilon\right)\). If \(\alpha\) is close to \(\frac{1}{2}\), this gives a new uniform lower bound \(\frac{5}{36}-\epsilon\) for the spectral gaps of Weil-Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of "second order cancellation". This is a joint work with Yuxin He and Yuhao Xue. ⚠️ Please note that the seminar will exceptionally take place at 08:00 EDT / 13:00 BST / 14:00 CEST. The talk will be recorded for the participants who cannot attend.
May 18	Malik Tuerkoen (UC Irvine)
	Concavity Properties of Dirichlet Eigenfunctions in Hyperbolic Space
	View Abstract On convex domains in \(R^n\) and \(S^n\), the first Dirichlet eigenfunction is known to be log concave, a fact that is crucial to estimate the spectral gap, which is the difference between the second and first Dirichlet eigenvalue. It is known that the first Dirichlet eigenfunction is in general not log-concave for convex domains in \(H^n\). I will discuss concavity estimates on horoconvex domains in hyperbolic space - which are domains whose boundaries second fundamental form is greater than 1 - which yield new spectral-gap bounds in H^n. In doing so, we resolve a conjecture by Nguyen, Stancu and Wei. This is based on joint work with G. Khan and on joint work with S. Saha and G. Khan.
June 1	Sugata Mondal (University of Reading)
	Small eigenvalues and their stability
	View Abstract After briefly recalling some history around small (or exceptional) eigenvalues, I will discuss the recent renewed interest in these eigenvalues from the perspective of their stability under finite Riemannian coverings.
June 8	Elena Kim (MIT)
	Polynomial bounds for eigenfunctions and eigenvalues on random covers of hyperbolic surfaces
	View Abstract Let \(X\) be a compact connected orientable hyperbolic surface and let \(X_n\) be a degree \(n\) random cover. We show that, with high probability, the \(L^\infty\) norm of every Laplace eigenfunction on \(X_n\) with bounded eigenvalue decays polynomially in \(n\). Using similar methods, we also show that, with high probability, the distribution of eigenvalues of the Laplacian on \(X_n\) converges to the spectral measure of the hyperbolic plane with polynomially decaying error. Our proof relies on the Selberg pre-trace formula and a variant of the polynomial method. This is joint work with Zhongkai Tao.