Program | Spectral Geometry in the Clouds

Program of the seminar for the current term

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.