Higher torsion invariants generalize the notion of the classical Reidemeister torsion. While the Reidemeister torsion is a tool for distinguishing between finite CW-complexes that are homotopy equivalent but not homeomorphic, higher torsion can detect bundles of smooth compact manifolds that are fiberwise homotopy equivalent (or even fiberwise homeomorphic), but have different smooth structures. In recent years various constructions of higher torsion invariants appeared including the higher analytical torsion of Bismut and Lott and Morse-theoretical construction of Igusa and Klein. The talk will present a construction of higher torsion that is based on the machinery of homotopy theory and some of its applications. The talk is based on joint work with W. Dorabiala, J. Klein and B. Williams.

In the second part I will outline the proof of rational convexity of singular Lagrangian inclusions with a Whitney umbrella. The proof consists of finding a pair of functions that satisfy certain properties. These functions are constructed as deformation of the standard symplectic structure on $C^2$.

In the first part of the talk, we recall the well-known correspondence between semi-groups and stochastic processes both in the commutative and non-commutative settings. We explain how this correspondence can be used to develop analysis on singular spaces such as fractals. Then, we proceed to a survey (including our results) of dilation theorems in operator algebras and how these theorems appear in the study of non-commutative stochastic processes.

We resume next week.

TBA

This is the third part of my talk on the local index formula. In this talk, I give two classical examples, the circle $\mathbb{T}$ and the torus $\mathbb{T}^2$ as spectral triples to demonstrate the use of the local index formula in the odd case and the even case, respectively.