# Stasheff J. Higher holonomy. Representations up to homotopy (abstract)

**Speaker**: Jim Stasheff

**Title**: Higher holonomy. Representations up to homotopy
**Abstract**:

Given a connection for a smooth vector bundle [math]\displaystyle{ p\colon E\to M }[/math], parallel transport with respect to smooth paths in the base space [math]\displaystyle{ M }[/math] provides a correspondence between smooth vector bundles with flat connection on [math]\displaystyle{ M }[/math] and representations of [math]\displaystyle{ \pi_1(M) }[/math]. Based in part on earlier groundbreaking work of K.T. Chen, recently this correspondence has been enhanced to the level of smooth paths (not homotopy classes) in the base space [math]\displaystyle{ M }[/math] and differential graded vector bundles with generalized flat connections.

Video

**Slides**: StasheffAMVconf2021slides.pdf

**Event:** Diffieties, Cohomological Physics, and Other Animals, 13-17 December 2021, Moscow.*Alexandre Vinogradov Memorial Conference.*