Mathematical Sciences: Geometric, Analytic, and Dynamical Properties of Real Submanifolds and CR Structures

9504452韦伯斯特拟议的研究是在几个复变量的领域；更具体地说，它处理复流形的实子流形。在该方案中强调了两个主题：研究者引入了双值反射的概念，以努力理解位于复代数曲线中的实代数曲线(因此双值反射的概念推广了共轭的概念)；抽象的Cauchy-Riemann结构的嵌入问题将被进一步探讨，重点是寻找可嵌入和不可嵌入的Cauchy-Riemann结构之间共享的那些性质。对柯西-黎曼结构的研究可以被认为是试图理解位于复杂空间中的某些真实表面；强调真实几何和复杂几何之间的相互作用。其结果是，为了更好地理解真实对象，通常有必要引入复杂变量。

9504452 Webster The proposed research is in the area of several complex variables; more specifically, it deals with real submanifolds of complex manifolds. Two topics are emphasized in the proposal: the investigator introduces a concept of double-valued-reflection in an effort to understand real algebraic curves sitting inside complex algebraic curves (thus the notion of double-valued-reflection generalizes that of conjugation); the embedding problem of abstract Cauchy-Riemann structures is to be pursued further, focusing on finding those properties shared between embeddable and non-embeddable Cauchy-Riemann structures. The study of Cauchy-Riemann structures can be thought of as an attempt to understand certain real surfaces sitting inside complex spaces; emphasizes the interplay between real and complex geometries. The upshot is that to understand a real object better it is often necessary to introduce complex variables.