Operator Algebras Seminar: The Higher Index of the de Rham Operator under Proper Cocompact Actions by a Discrete Group

Speaker: 向少聪 (华东师范大学)

Time: 2026-03-13 13:00-14:00
Location: 1402, SIMIS

Abstract:

In this talk, we discuss the higher index of the de Rham operator $d+d^*$ on manifolds equipped with a proper and cocompact action by a discrete group $\Gamma$. Motivated by Witten deformation, we indicate how the $K$-homology class of $d+d^*$ can be localized—within the $K$-theory of the group $C^*$-algebra—to neighborhoods of the zero set of a $\Gamma$-equivariant Morse-type vector field. We then apply this localization result to compute the higher index. Finally, we will discuss several concrete examples to illustrate these localized higher index formulas.