CR Manifolds and Singular Integrals

许多现代复分析都与切向柯西-黎曼（CR）算子的研究有关。 这个算子是通过将复欧几里得空间中的经典柯西-黎曼算子限制到一个超曲面或一个适当的称为CR流形的流形类而产生的。 这个运算符是不可逆的。 因此，为了理解它，我们必须理解一个相关的奇异积分算子，Szego投影算子，它是流形上平方可积函数空间到相关切CR算子零空间的正交投影。Szego投影对于有限型伪凸域的边界是比较好理解的。如果放松有限型假设，我们所知相对较少，如果放松伪凸性假设，我们所知就更少了。 系统地探索这些背景是当前项目的目标。 更具体地说，Halfpap和她的合作者将分析与几类超曲面和CR流形的Szego投影相关的积分核的显式表达式。 目标是1）确定Szego核的奇点的位置和大小，2）确定这些奇点如何与底层流形的几何相连接，以及3）理解投影算子本身的映射性质。一般来说，偏微分方程的研究是很重要的，因为偏微分方程给出了一种描述变化量之间关系的语言;一些模型热流，一些波传播，还有一些本身就是简单的研究对象。 对于一般的偏微分方程，出现了几个基本问题：可以解决未知函数的给定数据？解决方案是否独一无二？ 如果已知数据具有某些特殊属性（例如，光滑性或平方可积性）关于解可以说些什么？通常人们会寻求一个所谓的“基本解”--一个核心函数，人们可以对数据进行积分以获得原始PDE的解。因此，将解的性质与数据的性质相关联的定理需要关于相关联的积分核函数的详细信息。通常，核函数具有奇异性。 在这些情况下，得到一个奇异积分算子。这是当前项目所处的更大背景，因此当前项目的结果对偏微分方程和奇异积分算子出现的其他领域具有更广泛的影响。这个数学研究项目也有可能增加数学的多样性; Halfpap是一个积极的研究生顾问（以及有效的教师和本科生导师）在一所大学服务于一个大农村国家。

Much of modern complex analysis is concerned with the study of the tangential Cauchy-Riemann (CR) operator. This operator arises by restricting the classical Cauchy-Riemann operator in complex Euclidean space to a hypersurface or to an appropriate class of manifolds called CR manifolds. This operator is not invertible. Thus to understand it, we must understand a related singular integral operator, the Szego projection operator, which is the orthogonal projection of the space of square-integrable functions on a manifold onto the null space of the associated tangential CR operator. The Szego projection is relatively well-understood for boundaries of pseudoconvex domains of finite type. Comparatively little is known if one relaxes the finite-type hypothesis, and even less is known if one relaxes the pseudoconvexity hypothesis. Exploring these contexts systematically is the goal of the current project. More specifically, Halfpap and her collaborators will analyze explicit expressions for the integral kernel associated with the Szego projection for several classes of hypersurfaces and CR manifolds. The goals are 1) to determine the locations and sizes of the singularities of the Szego kernel, 2) to determine how these are connected to the geometry of the underlying manifold, and 3) to understand the mapping properties of the projection operator itself. In general, the study of partial differential equations (PDEs) is important because PDEs give a language for describing relationships among changing quantities; some model heat flow, some wave propagation, and some are simply objects of study in their own right. For a general PDE, several fundamental questions arise: Can one solve for the unknown function in terms of the given data? Is the solution unique? If the data are known to have certain special properties (e.g., smoothness or square-integrability) what can be said about the solution? Often one seeks a so-called ``fundamental solution" -- a kernel function against which one may integrate the data to obtain a solution of the original PDE. Theorems relating properties of the solution to those of the data thus require detailed information about the associated integral kernel function. Frequently, the kernel function has singularities. In these cases one obtains a singular integral operator. This is the larger context in which the current project is situated, and thus the results of the current project have broader implications for other areas in which PDEs and singular integral operators arise. This mathematics research project also has the potential to increase diversity within mathematics; Halfpap is an active advisor of graduate students (as well as an effective teacher and mentor of undergraduates) at a university serving a largely-rural state.