A major current in analysis and topology for the last decade has been the generalization of constructions and results in complex algebraic geometry to the almost complex, symplectic category. In recent work, Dr. Epstein has shown that the dbar-Neumann boundary condition has a symplectic analogue for Spin-C manifolds with contact boundary. This work is based on the notion of tame Fredholm pairs of projectors. Dr. Epstein will further explore this category and these boundary value problems, to obtain explicit formulae for the index of the Spin-C Dirac operator and gluing formulae under various convexity conditions. He will also use this framework to investigate the unmodified dbar-Neumann problem on a strictly pseudoconvex symplectic manifold, to see if there is a reasonable analogue of the Bergman projection, and if its range defines a useful analogue of holomorphic functions in the non-compact symplectic context.
In many mathematical and physical problems a question of principal interest is: how many solutions does a partial differential equation have? For certain classes of equations, a partial answer to this question, called the "index" of the equation, can be provided that does not depend on the details of equation. It is computed from the geometry of the space on which the solutions are defined. Dr. Epstein's work is broadly directly toward clarifying the relationship between this counting procedure and which aspects of the geometry of the underlying space are important determinants of the final result. One such approach to this problem involves cutting the space into parts and describing how the indices of the parts can be combined to compute the index of the whole.
The research supported by this grant was centered on the analysis of partial differential equations. This type of equation is used to model most physical phenomena. The equations I have studied model electromagnetic waves, the changes in the distribution of genotypes within a population, as well as abstract mathematical objects like complex- or CR-manifolds. My most abstract work concerns deformations of CR-structures on 3-dimensional contact manifolds. Broadly speaking this concerns the behavior of the space of solutions to a class of equation as the coefficients of the equation are varied. The equations I studied arise quite naturally in complex geometry on the boundary of complex surfaces. Even if the underlying geometric structure of the boundary is fixed, the space of such equations is infinite dimensional, and most equations in this class are pathological in that they do not have sufficiently many solutions. A basic question is whether or not a sequence of NON-pathological equations could have a pathological limit. I showed that this cannot happen by defining an invariant, called the relative index, and showing that it has an upper bound within the deformation space of CR-structures on a fixed compact 3-dimensional contact manifold. This constitutes a major step in a decades long effort of myself and other researchers to understand this and related phenomenon. A second major focus of my work concerns the representation of solutions to the time-harmonic Maxwell equations, and the application of such representations to the development of numerical methods for their fast and accurate solution. This work was done jointly with Prof. Leslie Greengard and his research group at NYU. These equations describe many things of general scientific and technological interest, for example the scattering of radar waves off of an aircraft, and the interaction of radio signals with the integrated circuits in a cell phone, or with buildings on a city street. While this is a classical subject, prior to our work all known methods suffered from one of several problems: 1. low frequency breakdown, 2. spurious resonances, 3. numerically instability due to poor conditioning of the underlying equations. None of these difficulties is intrinsic to the problem at hand, but is a consequence of the choice of representation for the solution. We found a new representation that simultaneously avoids all of these problems. In the process of this work we also clarified a range of issues in electromagnetic theory connected to the presence of boundaries that, like a bagel, are not simply connected. Our more flexible way of viewing this question also lead to a variety of insights about and improvements to existing methods. This work is currently being implemented as open access computer codes. The third main focus of my work is on models for the evolution of the distribution of types within a population. These probabilistic models, which are Markov processes, were introduced in population genetics by Wright and Fisher, and Kimura. While the main applications of these models are in population genetics, they are nearly identical to models used in mathematical finance for pricing options and futures. These models are described by partial differential equations for which the existence of solutions and their properties are far from obvious. In a joint project with Prof. Rafe Mazzeo of Stanford, we establish the existence, uniqueness and regularity properties of solutions to a large class of equations, which includes those appearing in population genetics. The existence of these solutions and their properties in turn provide a foundation for the probabilistic models alluded to above. Over the course of this grant I have studied a range of problems with significant applications to the development of Mathematics, as well as to Physics, Technology, and Biology. Funds from the grant were used to provide partial support for several graduate students working in these fields. The results of my research have been disseminated in the form of research papers, books, encyclopedia articles, public lectures and open source software.
