Structure of Mappings in Several Complex Variables and Cauchy-Riemann Geometry

不变几何的研究是数学的一个重要组成部分。不同类型的不变几何出现在不同的数学领域，例如，在几个复杂变量（SCV），偏微分方程（PDE），以及代数，复杂和微分几何。在这个研究项目中，首席研究员Peter Ebenfelt将研究SCV研究中出现的一种特殊几何形状，这种几何形状与量子场论和弦理论等数学物理领域的当代主题以及系统工程和控制理论等领域的应用也有很深的联系。本研究所需的工具来自数学的各个不同领域，如复分析、偏微分方程和微分几何，本研究中开发的技术和工具也影响着这些领域。Ebenfelt希望这个项目也能为研究生和博士后提供有趣的研究课题。从项目中产生的研讨会活动应该对学生和其他研究人员都有激励作用。Peter Ebenfelt的这个数学研究项目的目标是研究复杂变异体（更一般地说，是Cauchy-Riemann （CR）结构）中的一般实子流形及其映射的几何、解析和代数方面。Ebenfelt将考虑有关CR图的存在性、唯一性和规律性的问题，以及与本研究相关的相关问题。他将考虑Levi非简并超曲面到高维非简并超二次曲面的CR映射的背景。本研究将加深我们对构成Levi非简并超曲面理论中平面模型的超二次曲面的CR子流形结构的理解。所提出的工作还应该提供对这些超曲面的局部CR几何（原则上完全编码在CR曲率张量中）如何影响其CR映射的各种性质的见解，例如非简并性和刚性的概念。Ebenfelt还将通过研究将CR映射定义为射流束上的奇异Pfaffian系统的系统的扩展，继续他在无限型一般子流形之间的CR映射的研究。他将特别关注于理解最近发现的现象，即对于无限型超曲面，生物全纯、形式和光滑CR等价分类是不同的。他还将研究一个关于局部自同构的有限射流确定的猜想。这些研究有望揭示无限型流形之间CR映射的性质，并导致更好地理解在这种情况下产生的Pfaffian系统。Ebenfelt还将继续研究复2空间无限型超曲面的标准形式。最后，Ebenfelt将研究Kähler流形上单位圆束上的CR不变量，特别是三维情况下的Cartan脐带张量以及Bergman和Szegö核的展开中产生的CR不变量。他打算发展一般三维CR流形上脐带点检测的新方法，并解决关于嵌入复2空间的紧致CR流形上脐带点存在性的开放性问题。

The study of invariant geometries is an important part of mathematics. Different types of invariant geometries arise in different areas of mathematics, e.g., in several complex variables (SCV), partial differential equations (PDE), and algebraic, complex, and differential geometry. In this research project, the principal investigator Peter Ebenfelt will investigate a particular geometry that arises in the study of SCV, and which has also deep connections with contemporary topics in mathematical physics such as quantum field theory and string theory, as well as applications in, e.g., systems engineering and control theory. Tools that are needed for this study come from a variety of different areas in mathematics, such as complex analysis, PDE, and differential geometry, and the techniques and tools developed in this study influence these areas as well. Ebenfelt expects that the project will also provide interesting research topics for graduate students and postdocs. The seminar activity that results from the project should be stimulating for both students and other researchers.The goal of this mathematics research project by Peter Ebenfelt is to study geometric, analytic, and algebraic aspects of generic real submanifolds in complex varieties (more generally, of Cauchy-Riemann (CR) structures) and their mappings. Ebenfelt will consider questions regarding existence, uniqueness, and regularity of CR maps, as well as related questions that arise in connection with this study. He will consider the context of CR maps of a Levi nondegenerate hypersurface into a nondegenerate hyperquadric of higher dimension. This study will enhance our understanding of the CR submanifold structure of the hyperquadrics, which constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The proposed work should also provide insight into how the local CR geometry of such hypersurfaces (in principle completely encoded in the CR curvature tensor) affects various properties, such as notions of nondegeneracy and rigidity, of their CR maps. Ebenfelt will also continue his study of CR maps between generic submanifolds of infinite type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. He will, in particular, focus on understanding the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are different. He will also study a conjecture regarding finite jet determination of local automorphisms. These investigations are expected to shed new light on the nature of CR maps between infinite type manifolds, and lead to a better understanding of the Pfaffian systems arising in this context. Ebenfelt will also continue his work on normal forms for infinite type hypersurfaces in complex 2-space. Finally, Ebenfelt will study CR invariants on unit circle bundles over Kähler manifolds, specifically Cartan's umbilical tensor in the 3-dimensional case and those arising in the expansions of the Bergman and Szegö kernels. He intends to develop new methods for detecting umbilical points on general 3-dimensional CR manifolds, and resolve an open problem regarding existence of umbilical points on compact CR manifolds embedded in complex 2-space.