Speaker: Professor Yannick Sire (Johns Hopkins University)

Time: 2024年9月20日16:00

Venue: 数学科学学院2楼报告厅（海纳苑2幢210室）

Abstract: I will describe various upper and lower bounds on the spectrum of the Laplace-Beltrami (and associatedl Schrödinger) operator on Riemannian manifolds. The upper bounds lead to some important results in spectral geometry establishing a link between the so-called conformal spectrum (and extremal metrics) and branched minimal immersions into Euclidean spheres. I will as well describe some estimates on spectral projections, which amount to understand some concentration properties of eigenfunctions. Derivation of those various bounds are aimed at understanding the geometry and topology (and dynamical properties) of manifolds in terms of spectral data.

I will then describe a conjecture by Yau on the first eigenvalue on minimal submanifolds of the sphere, l which is known only for some examples and still largely open. I will then present some recent results where l we improve quantitatively the best known lower bound (in the general case) of Choi and Wang of the mid 80’s. I will address some open problems and possible directions to generalize the results.

Profile: Yannick Sire is now a professor at Johns Hopkins University. He is a French citizen born in the Roussillon region of France. He did his undergraduate studies from 1997 to 2002 in the city of Toulouse, pursuing at the same time an engineering degree in applied mathematics at Institut National Des Sciences Appliquee's and ad degree in pure mathematics at Universite' Paul Sabatier. After obtaining a Master's degree, he started a Ph.D. at Universite' Paul Sabatier under the supervision of J. M. Roquejoffre and G.James, completed in 2005. He spent the years 2005-2007 as a Postdoctoral fellow at University of Texas at Austin, mentored by Luis Caffarelli. Beginning in 2007 he was hired at the University of Aix-Marseille in a tenured position.