Contact Geometry, Complex Analysis and Imaging

在过去的十年里，分析和拓扑学的一个主要趋势是将复杂代数几何的构造和结果推广到几乎复杂的辛范畴。在最近的工作中，Epstein博士已经证明了dbar-Neumann边界条件对于具有接触边界的自旋- c流形具有辛模拟。这项工作是基于驯服弗雷德霍姆对投影仪的概念。Dr. Epstein将进一步探索这一类和这些边值问题，得到Spin-C Dirac算子指标的显式公式和各种凸性条件下的胶合公式。他还将使用这个框架来研究严格伪凸辛流形上的未修改dbar-Neumann问题，看看是否存在Bergman投影的合理模拟，以及它的范围是否定义了非紧辛环境下全纯函数的有用模拟。在许多数学和物理问题中，一个主要的问题是：一个偏微分方程有多少个解？对于某些类型的方程，可以给出这个问题的部分答案，称为方程的“指标”，它不依赖于方程的细节。它是从定义解的空间的几何构造中计算出来的。爱泼斯坦博士的工作基本上是直接为了澄清这种计数过程与底层空间几何的哪些方面是最终结果的重要决定因素之间的关系。解决这个问题的一种方法是将空间分割成各个部分，并描述如何将各个部分的指数组合起来计算整体的指数。

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.