Higher Structure Conferences higherstructurechina V (Rev #4, changes)

(1) Natural higher structures in geometry and topology

Abstract: In this talk, I will review some basics of topological spaces, smooth manifolds, and vector bundles and the role of cocycle conditions in their construction. Then I will explain how violations of the cocycle conditions lead to natural higher structures in geometry and topology:

orientation structures, spin structure and string structure. The talk will end with a 30-year outstanding question about Dirac operators on loop spaces.

(2) Gerbes, D-branes and index theory

Abstract: In this second talk, I will explain another notion of higher structures: gerbes, and its role in the study of D-branes. Then I will give a mathematical definition of D-branes and its relation to twisted version of Atiyah-Singer index theorem.

An introduction to index theory of invariant Dirac operators

In the first hour I will introduce Dirac operators and some properties, on manifolds and manifolds with group actions, and definition of higher index. In the second hour I will talk about some variations of indices and their connections to representation theory.