Spectral Geometry in the Clouds
Intergalactic seminar
on spectral theory and geometric analysis
on spectral theory and geometric analysis
| Home | Program | Useful Links |
| October 6 | Lawford Hatcher (Indiana University, Bloomington) |
| Comparing Dirichlet and Neumann Laplace spectra | |
View AbstractUnderstanding 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. |
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| October 13 | Lukas Bundrock (University of Alabama) |
| Spectral Optimization in Exterior Domains: Robin and Steklov Problems | |
View AbstractThe 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. |
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| October 20 | Dorin Bucur (Université de Savoie) |
| The geometric size of the fundamental gap | |
View AbstractThe 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. |
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| October 27 | Guenda Palmirotta (Universität Paderborn) |
| Patterson-Sullivan distributions of hyperbolic surfaces | |
View AbstractThere 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). |
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| November 3 | Denis Vinokurov (Université de Montréal) |
| Maximizing Laplace eigenvalues with density in higher dimensions | |
View AbstractWe 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). |
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| November 10 | Maurizia Rossi (University of Milano-Bicocca) |
| The nodal length of random spherical harmonics | |
View AbstractIn 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.) |
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| November 17 | Jared Wunsch (Northwestern University) |
| Gutzwiller trace formula for singular potentials | |
View AbstractThe 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. |
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| November 24 | Romain Speciel (Stanford University) |
| Commutativity properties of the Dirichlet-to-Neumann map | |
View AbstractThe 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]. |
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| December 1 | Joe Thomas (Durham University) |
| Spectral gap for random hyperbolic surfaces | |
View AbstractThe 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. |
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| December 8 | Iosif Polterovich (Université de Montréal) |
| Courant, Bézout, and topological persistence | |
View AbstractFunctions 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. |
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| December 15 | Daniel Stern (Cornell University) |
| Electrostatics and minimal surface doublings | |
View AbstractI'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. |
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| January 19 | INI joint seminar |
| Eugenia Malinnikova (Stanford University) | |
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| January 26 | TBA (Affiliation) |
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| February 9 | TBA (Affiliation) |
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| February 23 | TBA (Affiliation) |
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| March 2 | TBA (Affiliation) |
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| March 9 | TBA (Affiliation) |
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| March 16 | TBA (Affiliation) |
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| April 27 | Yunhui Wu (Tsinghua University) |
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| May 4 | TBA (Affiliation) |
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| May 18 | TBA (Affiliation) |
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| May 25 | TBA (Affiliation) |
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| June 1 | TBA (Affiliation) |
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| June 8 | TBA (Affiliation) |
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| June 15 | TBA (Affiliation) |
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