This talk will present an extension of the notion of spectral triples that accommodate several types of hypoelliptic differential operators on manifolds with boundary. A so-called higher-order relative spectral triples give rise to a relative $K$-homology class using the bounded transform which makes it feasible to calculate the boundary map in $K$-homology of said class. When we consider the class constructed from an elliptic differential operator on a compact smooth manifold with boundary, the image of the $K$-homology boundary map can be explicitly described to obtain a generalization of the Baum-Douglas-Taylor index theorem. This also provides a generalization of Boutet de Monvel's index theorem and a Poincaré-dual description of Aityah-Bott's obstruction of the existence of elliptic boundary conditions.

Bio: Magnus Fries is a Swedish mathematician. Since 2021 Magnus has been doing his PhD at Lund University (Sweden) with supervisor Magnus Goffeng. Magnus' work is mainly in index theory using tools from non-commutative geometry with an angle to boundary conditions for differential operators.