Program | Spectral Geometry in the Clouds

Program of the seminar for the current term

October 19	Matthew Colbrook (University of Cambridge)
	Beyond the circle: Schiffer's conjecture and computation
	View Abstract When does the behaviour of a single eigenfunction determine the shape of a domain? Schiffer’s conjecture predicts that a bounded, simply connected planar domain must be a disc if it admits a nonconstant Neumann eigenfunction that is constant on the boundary. I will present joint work with George Stepaniants disproving this conjecture by constructing a noncircular domain with real-analytic boundary. The same example disproves the planar Pompeiu conjecture, connecting spectral geometry to a problem about averages over translated and rotated domains dating back to 1929. The challenge is not merely to find a convincing numerical picture, but to prove that an exact domain and eigenfunction exist. I will explain how conformal mapping, spectral analysis and a contraction argument turn a numerical candidate into a theorem. The crucial step is to control everything a finite computation leaves out, including the infinitely many omitted modes, while rigorously accounting for numerical error. After finishing this paper, I have started to use AI to assist me in spectral problems. I will end the talk by drawing on recent work on eigenvalue certification and the conjectures of Forsythe and Kenig to discuss the changing role of computers and AI in mathematical discovery. These examples range from assistance with code to the interactive discovery of proof mechanisms and formal verification in Lean. I will also discuss our initiative “AIM, explained” for the human digestion of AI proofs that have appeared in the last month.
October 26	Iosif Polterovich Université de Montréal
	The Faber-Krahn position, convexity, and Gaussian measure inequalities Special time: 8am PST, 11am EST, 3pm GMT, 4pm CET
	View Abstract We say that a convex body is in Faber–Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. We prove the uniqueness of this position up to orthogonal transformations, answering a question of Schmuckenschläger from 2011. The proof combines inequalities for Gaussian measures with the probabilistic interpretation of the fundamental tone of a domain. Other spectral applications of Gaussian measure inequalities will also be discussed, including eigenvalue estimates for intersections and Minkowski sums of convex bodies. The talk is based on a joint work with Dmitry Faifman.
November 2	TBC

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November 9	TBC

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November 16	Andrii Khrabustovskii University of Hradec Králové

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November 23	TBC

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November 30	Daniel Platt Imperial College

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December 7	Alex Cohen New York University

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