Colloquium: Index and secondary Z_2- valued index invariants in geometry and physics.

Abstract: I will recall the notions of the index of an elliptic operator and the spectral flow of a family of such operators, and discuss various applications of these invariants, including their role in the study of conductance in solid-state physics. I then turn to the situation in which a symmetry forces these invariants to vanish, and explain how secondary, Z_2​-valued analogues of the index and spectral flow can be defined in such cases. I will discuss properties of these secondary invariants for geometrically defined operators — in particular, analogues of several classical theorems relating the index and the spectral flow to their secondary counterparts. Finally, I will explain how the secondary invariants yield information about the conductance of certain materials. (Based on joint works with Ahmad Reza Haj Saeedi Sadegh)

Host: Yanli Song