Brief Introduction to Probability
Prof. Sengupta's research work has centered around the following
topics: (1) mathematical problems connected with geometric
quantum field theories, (2) problems involving infinite-dimensional
stochastic analysis, and (3) study of the Segal-Bargmann transform.
The mathematical ideas involved include probability theory and
stochastics, differential geometry and topology, and symplectic
structures. Some of the motivation for the problems comes from
quantum physics.
Sengupta's work has involved the construction of a probability
measure on an infinite dimensional non-linear space whose points
represent geomeric objects called connections. The study of this
measure is motivated by quantum gauge theory in two dimensions. The
construction involves an infinite-dimensional Gaussian measure
conditioned to satisfy certain constraints imposed by topology and
geometry.
As a limiting case of the measure described above, one obtains a
measure on the moduli space of flat connections over a surface.
This space is of interest from a variety of points of view,
including topology and algebraic geometry. Sengupta has been
studying primarily the symplectic structure on this moduli space
(when the underlying surface is oriented).
The Chern-Simons integral is an object which arises again from a
geometric quantum field theory and it was shown by Witten that this
integral provides a way to determine topological invariants
associated with knots in three dimensional manifolds. In work done
jointly with Professor S. Albeverio (Bonn), Sengupta has also
constructed a rigrorously meaningful version of the Chern-Simons
functional integral as an infinite-dimensional distribution.
Further work needs to be done to evaluate this distribution on
functions of interest, thereby establishing the connection with
topology in a mathematically rigorous way.
Sengupta is also interested in infinite-dimensional distribution
theory (white noise analysis) and, along with Professors Cochran
and Kuo (both of LSU), has investigated general classes of such
distributions and the effect of a standard transform called the
S-transform on such distributions.
Sengupta has also been investigating, mainly in collaboration
with Professor Brian Hall (Notre Dame), the Segal-Bargmann
transform, a unitary isomorphism connecting spaces of
square-integrable functions with spaces of holomorphic
square-integrable functions, in a Gaussian background.
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