Heat kernel weighted L^2 spaces (Driver)

 

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6th Cornell Probability Summer School (July 19 - 30, 2010)

http://www.math.cornell.edu/~cpss/

Instructor:        Bruce Driver

Title:                   Heat kernel weighted L2 spaces

 Description:     We will discuss L2 -- 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:

  1.  Preliminary measure theoretic and holomorphic function theory for Banach spaces.

  2.  Gaussian measures an Banach spaces including;

    1. The Cameron-Martin space and theorem and

    2. heat equation interpretations.

  3. Gaussian processes as Gaussian measures along with the informal path integral description of Brownian motion and how to interpret it rigorously.

  4. A detour into path integral quantization;

    1. basic idea in for finite dimensional configurations spaces

    2. Hilbert space support properties of Gaussian measures

    3. applications to (not) understanding quantum field theories.

  5. Heat kernel smoothing, the Segal - Bargmann transforms, and the Kakutani-Itô-Fock space isomorphisms.

  6. Applications to canonical quantization of Yang-Mills in 1+1 dimensions leading to Gross' generalization of the Kakutani-Itô-Fock space isomorphisms and Hall's generalization of the Segal Bargmann transform.

  7. Topics that might of been covered but alas were not;

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

    2. 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.

    3. Interpretation of path integrals on manifolds.

Recommended background reading

Although I will try to keep this course as self-contained as possible, you may find it helpful to look over the following material ahead of time.

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

  2.  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:

  1. Gaussian measure spaces review.

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

  3. Quantize Yang-Mills Fields in 1 space and 1 time dimension.

  4. Hall's isometry theorem for compact Lie groups.

  5. The Taylor isomorphism for Lie groups

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

 

http://www.math.ucsd.edu/~bdriver/Cornell Summer Notes 2010/index.htm

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Last modified on Wednesday, 16 June 2010 05:59 PM.