Speakers

Refinements in Local Index Theory

Abstract: The Atiyah-Singer index theorem is one of the great pillars in global analysis connecting the fields of analysis, topology, and geometry. To date, there are various approaches to the subject of index theory, local index theory having its focus on deriving characteristic forms from the local coefficients of a geometric elliptic operator. For a spin Riemannian manifold, the natural choice of operator is the Dirac operator on spinors. In this talk we will discuss recent refinements on the local geometry of the Dirac operator as well as the local geometry of a spin Riemannian manifold.

On mod two index of odd symmetric operators on non-compact manifolds

Abstract: In this joint work Maxim Braverman, we investigate elliptic operators with a symmetry that forces their index to vanish. We study the secondary index, defined modulo 2. We examine Callias-type operators with this symmetry on non-compact manifolds and establish mod 2 versions of the Gromov-Lawson relative index theorem, the Callias index theorem, and the Boutet de Monvel’s index theorem for Toeplitz operators.

Mackey Analogy and Higher Orbital Integrals

Abstract: The Mackey analogy is a phenomenon in representation theory that connects the space of representations of a reductive group with the space of representations of an associated motion group. In 2020, Nigel Higson and I successfully constructed an embedding, now called the Mackey embedding, between reduced group C*-algebra of the motion group and the reduced group C*-algebra of the complex reductive group.

Now, in a joint work Yanli Song and Xiang Tang, we attempt to bring the Mackey analogy to the higher orbital integrals of reductive groups and their associated motion groups. I will first present some results on the orbital integrals and their relations via the Mackey analogy. I will then present some formulas describing the higher orbital integrals for both the reductive group and the motion group.

Hilsum bordism and relative constructions in unbounded KK-theory

Abstract: Many cycles in Kasparov's KK-theory are obtained from unbounded operators. Prototypical examples include the cycles associated to geometrically defined operators on a manifold (e.g., the signature operator, the spin^c Dirac operator, etc). Baaj and Julg defined the notion of an unbounded cycle in KK-theory and more recently, Hilsum defined the notion of a bordism in the context of unbounded KK-theory. Hilsum's definition is based on operators associated to manifolds with boundary.

In joint work with Magnus Goffeng and Bram Mesland, we defined an abelian group which is essentially unbounded KK-cycles modulo Hilsum's notion of bordism. This group maps to the standard Kasparov group via the bounded transform and in the commutative case can be related to the geometric model for K-homology due to Baum and Douglas. In particular, these KK-bordisms allow for relative constructions that are similar to (but more analytic than) ones developed in previous joint work with Magnus Goffeng in the context of geometric K-homology. I will discuss such relative constructions starting with those in geometric K-homology and then moving to KK-bordisms.