Magnus Fries: An obstruction of elliptic boundary conditions and a search for eigenvalue inequalities

Abstract: The Boutet de Monvel index theorem is a generalization of the Toplitz
index theorem and closely relates to Noether's original example
regarding Fredholm index. This index theorem was proven by
Baum-Douglas-Taylor using the boundary map in \(K\)-homology, and we
have proven that the boundary map provides an obstruction of existence
of elliptic boundary conditions associated to an elliptic differential
operator. For instance, there are no elliptic boundary conditions we can
impose on the spin Dirac operator on a strictly pseudoconvex domain,
otherwise the index in the Boutet de Monvel index theorem would always
be zero. In contrast, the de Rham Dirac operator has a natural choice of
elliptic boundary conditions which lies behind Rohleder's recent results
regarding the Friedlander inequality. Our hope is to find other suitable
differential complexes to obtain new eigenvalue inequalities using
similar methods.