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 ℝN. 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 ℝN, 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) |
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| November 10 | Maurizia Rossi (University of Milano-Bicocca) |
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| November 17 | Jared Wunsch (Northwestern University) |
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| November 24 | Romain Speciel (Affiliation) |
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| December 1 | Joe Thomas (Durham University) |
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| December 8 | Iosif Polterovich (Université de Montréal) |
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| December 15 | Daniel Stern (Cornell University) |
| TBA | |
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The program for the second term will be made available later.