Mathematics Colloquium, Fall 2005

Noncommutative Fourier transform, Chen's iterated integrals and path integrals

The usual exponential functions exp, x=(x₁, ..., x_n), depend on the vector a of exponents. Various ways of lifting such a function to noncommutative variables x_i (which can be, say, indeterminate matrices) depend on a choice of a path joining 0 and a in the phase space. This lifting can be seen as the generating function of the iterated integrals of Chen. Since there are as many noncommutative exponentials as there are paths, one is led to considering a "noncommutative Fourier transform" which would identify an appropriate space of functions of noncommutative variables and the space of ordinary functions (or measures) on the space of paths. The talk will discuss various aspects of such (conjectural) transform.

Thursday, December 1, 2005, 245 Altgeld Hall, 4:00 p.m.