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The Campbell-Baker-Hausdorff Formula, A Graphical Calculus, and Iterated Integrals.
Geometry/Topology| Speaker: | Vinay Kathotia, Mathematics, UC Davis |
| Location: | 693 Kerr |
| Start time: | Wed, Jun 2 1999, 4:10PM |
For two non-commuting variables X and Y we do not have exp(X)exp(Y) = exp(X+Y) but the Campbell-Baker-Hausdorff (CBH) Formula does give us exp(X)exp(Y) = exp(C), where C = X + Y + (1/2)[X,Y] + (1/12)[X,[X,Y]] + ... is a (formal) weighted sum of X, Y, and their commutators. There is no known `closed' formula that unambiguously gives the weight for a fixed commutator expression in C.
Less than two years ago M. Kontsevich (in a paper on deformation quantization) introduced certain weighted graphs that encode the CBH formula and may lead to a better understanding of it. This talk will outline Kontsevich's simple yet mysterious construction and relate it to the CBH formula.
The topological import of Kontsevich's construction is unclear but we will attempt to address the issue. In particular, his weights are iterated integrals and we will relate these to iterated integrals introduced by K.T. Chen for problems in homotopy theory and parallel transport.