From: Brenda Sue Cook (brenda@math.missouri.edu)
Date: Thu Sep 20 2007 - 14:02:08 CDT
Modern Analysis Seminar
Matt Cecil (University of Connecticut)
The Taylor Map on Complex Path Groups
Abstract:
A well-known fact from elementary complex analysis is that a holomorphic
function on $\mathbb{C}$ is determined by its derivatives at the origin
via its everywhere convergent Taylor expansion. Given a Gaussian
measure
$\mu_t$, one can view this Taylor expansion as giving a Hilbert space
isomorphism from $\mathcal{H}L^2(\mu_t)$ to a space of ``derivatives at
the origin" endowed with an appropriate norm. This ``Taylor map" is one
part of the larger Fock space isomorphism which has a natural
interpretation in the context of quantum fields.
The Taylor map can be formulated in non-commutative settings (replace
$\mathbb{C}$ by a complex Lie group above) and in many infinite
dimensional settings as well. I will review these known results and
highlight the case of the Taylor map acting on holomorphic functions on
$\mathcal{W}(G)$, the path group of a connected complex Lie group, which
are square integrable with respect to an appropriate heat kernel
measure.
The topic offers a nice blend of probability, geometry, and analysis
and I
will attempt to present it as such.
312 Math Sci
2:00-2:50 p.m.
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