Everytopic Seminar

 February 29th, 2008, 1:40 pm – 3:00pm, Goldsmith 226



Title: Iterated integrals and chain homotopy

Speaker: Kiyoshi Igusa, Brandeis University



The purpose of this talk is to explain the basic definition

and properties of d-dimensional iterated integrals for d=1,2,3, in particular the case d=2.


(d=1): These are Chen's iterated integrals. I will explain how iterated integration

of a connection on a vector bundle gives the holonomy.

In the special case that I study, these holonomies are chain maps between chain complexes.


(d=2): 2-dimensional iterated integrals give ``chain homotopies.'' This is the main point of the lecture.


(d=3): in dimension 3 we get a degree 2 mapping between chain complexes whose boundary

is a chain homotopy. I.e., this is a ``higher homotopy.''


These integration formulae convert

Riemannian geometry into category theory and homological algebra.

It is a long story of which I am telling only one part.