Matthew S. Cecil

Visiting Assistant Professor

Ph.D., University of California-San Diego, 2006

M.S., Univerisyt of California-San Diego, 2003

Research Interests

I am interested in stochastic process and heat kernel analysis on Lie groups and manifolds, including infinite dimensional spaces such loop and path groups.  Stochastic process (like Brownian motion) give useful representations of solutions of heat equations, and properties of these solutions are intimately related to the underlying geometry.  I am primarily interested in function spaces given by these heat kernel measures and functional inequalities, so my research can be interpreted as a blend of probability, geometry, and functional analysis.

Much of my current research is on questions related to the Taylor map.  Briefly stated, this is an isometry between square integrable holomorphic functions on a complex space and a weighted space of derivatives at a point.  This is somewhat analogous to the classical Taylor expansion (hence the name), and has an interpretation in the Fock space representation of Quantum Mechanics.  Much of the motivation for my work comes from Physics.

Selected Publications

• The Taylor Map on Complex Path Groups, Journal of Functional Analysis, 254, pp. 318-367, 2008.
• Heat Kernel Measures on Loop and Path Groups, Infinite Dimensional Analysis, Quantum Probability, and Related Topics, Vol. 11, 2, pp. 135-156, June 2008.
• The Ricci Curvature of Finite Dimensional Approximations to Path and Loop Groups, To Appear in Bulletin des Sciences Mathematiques, 23 pgs.
• An Embedding of a Lie Group in the Universal Enveloping Algebra of its Lie Algebra, Preprint.

Please direct questions and comments to: cecil.12@nd.edu