Mikhail Karpukhin

Harmonic maps and eigenvalue optimization in higher dimensions

The study of sharp upper bounds for the Laplace eigenvalues on surfaces under the volume constraint is a classical problem of spectral geometry. It has surprising applications to counting negative eigenvalues of Schrodinger operator as well as the theory of harmonic maps to spheres. In the present talk, we will survey some recent advances in the field and discuss their generalizations to higher dimensional manifolds. Based on a joint work with D. Stern.