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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 |
Description
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.