Announcements Driver's Notes Reference Material
 
Instructor: Bruce
Driver
Title:
Heat kernel weighted L^{2} spaces
Description: We will discuss
L^{2}
 spaces relative to certain heat kernel measures. Our concentration will
be on describing these spaces in terms of their ``Taylor'' coefficients.
Here is the course outline:

Preliminary measure theoretic and holomorphic
function theory for Banach spaces.
 Gaussian measures an Banach spaces including;

The CameronMartin space and theorem and

heat equation interpretations.

Gaussian processes as Gaussian measures along with the
informal path integral description of Brownian motion and how to interpret
it rigorously.
 A detour into path integral quantization;

basic idea in for finite dimensional configurations
spaces

Hilbert space support properties of Gaussian measures

applications to (not) understanding quantum field
theories.

Heat kernel smoothing, the Segal  Bargmann transforms,
and the KakutaniItôFock space isomorphisms.

Applications to canonical quantization of YangMills in
1+1 dimensions leading to Gross' generalization of the KakutaniItôFock
space isomorphisms and Hall's generalization of the Segal Bargmann
transform.
 Topics that might of been covered but alas were not;

The Taylor isomorphism for arbitrary finite
dimensional Lie groups equipped with subelliptic heat kernel measures.

Extensions of parts of this theory to path and loop
groups, Hilbert  Schmidt groups (see Masha Gordina's lectures), and
infinite dimensional nilpotent Lie group  see the work of Matt Cecil
and Tai Melcher.

Interpretation of path integrals on manifolds.
Recommended background reading
Although I will try to keep this course as selfcontained as
possible, you may find it helpful to look over the following material ahead of
time.

See Part VIII  Banach Spaces and Probability in
Probability Tools with Examples.

See
Example 2.6 on p. 3 and Exercise 2.28 on p. 8 of
Curved Wiener space analysis.
These examples introduce you to the notion of a manifold
and more importantly a Lie group.
Proposed Course Outline (Not followed!)
The tentative topics for the 6 lectures are:

Gaussian measure spaces review.

The Kakutani  Ito  Segal  Bargmann transforms and the
relation to multiple Ito^integrals (Continuation of Masha Gordina's first talk.

Quantize YangMills Fields in 1 space and 1 time dimension.

Hall's isometry theorem for compact Lie groups.

The Taylor isomorphism for Lie groups

Extensions of the Theory  Give a survey of some or our
results in the infinite dimensional Heisenberg groups.
