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)
	TBA
	View Abstract TBA
March 2	Mitchell Taylor (ETH Zürich)
	TBA
	View Abstract TBA
March 9	INI joint seminar
	Alexander Strohmaier (Leibniz Universität Hannover)
	TBA
	View Abstract TBA
March 16	TBA (Affiliation)
	TBA
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April 27	Yunhui Wu (Tsinghua University)
	TBA
	View Abstract TBA ⚠️ 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 4	Sugata Mondal (University of Reading)
	TBA
	View Abstract TBA
May 18	Antonio Celentano (Università degli Studi di Napoli Federico II)
	Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane
	View Abstract TBA
May 25	TBA (Affiliation)
	TBA
	View Abstract TBA
June 1	TBA (Affiliation)
	TBA
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June 8	TBA (Affiliation)
	TBA
	View Abstract TBA
June 15	TBA (Affiliation)
	TBA
	View Abstract TBA