Contact Geometry, Complex Analysis and Imaging

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Abstract for "Contact geometry, complex analysis and imaging" DMS- 0203705 Dr. Epstein's research proposal contains projects in four distinct areas: (1) Invariants of contact structures, (2) Problems in CR-geometry, (3) Multi-parameter perturbation theory, (4) Magnetic resonance imaging. The proposed work in (1) will likely be done in collaboration with Richard Melrose. They intend to use their prior work on the Heisenberg calculus to find invariants of contact manifolds. A problem of particular interest is to use spectral flow invariants of Toeplitz operators to distinguish contact structures with homotopic hyperplane fields. The work in (2) is part of Dr. Epstein's ongoing project on embeddability of CR-manifolds, he proposes to consider CR-structures in dimensions greater than 3 to understand why some CR-functions are unstable and why some are stable under deformation of the underlying CR-structure. The project described in (3) concerns the problem of analytically parameterizing the eigenvalues and eigenspaces of a "self adjoint" analytic family of matrices. In broad terms Dr. Epstein would like to see what the modern theory of functions of several complex variables has to say about multi-parameter perturbation problems. Preliminary investigations show that real progress is now possible in this field. His main goal is to find reasonably simple and explicit analytic spaces on which the eigenvalues and eigenspaces of a self adjoint family are analytically parametrized. Finally (4) is an entirely new direction for Dr. Epstein's research. He plans to study the feasibility of magnetic resonance imaging using inhomogeneous background fields. Dr. Epstein's work is in the application of analysis and algebra to problems in geometry. A principal focus of his research is the geometry and analysis of contact manifolds. These arise is many contexts from several complex variables to differential geometry to topology to partial differential equations. Using the tools developed in collaboration with Richard Melrose, Dr. Epstein hopes to elucidate the analysis and geometry of these ubiquitous spaces. Dr. Epstein also plans to investigate the feasibility of magnetic resonance imaging using inhomogeneous background fields. This is largely beyond the capabilities of present day technology. If it proves feasible it would allow many new applications of MR imaging and should also lead to less expensive equipment. This is a multifaceted problem with significant mathematical and engineering aspects.