20 may 2018 - 26 may 2018 [open in google calendar]
The course is addresses to Ph.D. or post-doc students, and aims to introduce the attendants to various techniques to study the spectrum of relevant operators on manifolds. Motivations and applications come from problems in Geometric Analysis, and will be highlighted in the course.
Programme overview: (1) Basics: Laplace, Schrödinger and Dirac operators with special focus on noncompact manifolds, Geometry of submanifolds and immersions, Introduction to the spectral theory of compact minimal hypersurfaces
(2) Techniques to study the spectrum on noncompact manifolds: Rayleigh-Ritz and minmax, Radialization, Heat kernels and Green functions, decay estimates for embedded eigenvalues, Conditions for discreteness of spectra half lines in the essential spectra existence of spectral gaps
(3) Applications: Geometric implications to immersions, submanifold theory: index estimates and harmonic forms, spectral stability and topology of manifolds, Weak compactness of closed minimal hypersurfaces under spectral bounds, (Non-)existence results for geometric semilinear PDEs, e.g. Yamabe-type equations.
Introductory reading: Some spectral theory: Chapter 3 in Pigola S., Rigoli M. and Setti A., Vanishing and Finiteness Results in Geometric Analysis, Progress in Math. 266, 2008
Organizers: Nadine Grosse, Luciano Mari, Ben Sharp.
Speakers: Nadine Grosse, Luciano Mari, Ben Sharp.