Abstract: Nonabelian multiplicative integration on curves is a classical theory, going back to Volterra in the 19-th century. In differential geometry this operation can be interpreted as the holonomy of a connection along a curve.

This talk is about the 2-dimensional case. A rudimentary nonabelian multiplicative surface integration was known since the 1920´s (work of Schlesinger). I will present a much more sophisticated construction. My main result is a 3-dimensional nonabelian Stokes Theorem. This result is completely new; only a special case of it was predicted (without proof) in papers in mathematical physics.

The talk is fairly elementary, requiring only some knowledge of Lie groups and their Lie algebras. And there are many color pictures!

My motivation for this work has to do with a problem in twisted deformation quantization. I will say a few words about this at the end of the talk.

For full details see the lecture notes http://www.math.bgu.ac.il/~amyekut/lectures/multi-integ/notes.pdf or the preprint arXiv:1007.1250 at http://arix.org.