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
http://cvgmt.sns.it/event/461/