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

不变几何的研究是数学的重要组成部分。不同类型的不变几何出现在不同的数学领域，例如，在几个复变量(SCV)、偏微分方程(PDE)以及代数、复几何和微分几何中。在这个研究项目中，首席研究员彼得·艾本费尔特将研究一种在SCV研究中出现的特殊几何，它也与当代数学物理中的主题，如量子场论和弦理论，以及在系统工程和控制理论中的应用有着深刻的联系。这项研究所需要的工具来自数学中的各种不同领域，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到这些领域。艾本费尔特预计，该项目还将为研究生和博士后提供有趣的研究主题。该项目的研讨会活动对学生和其他研究人员都应该是有启发的。Peter Ebenfit的这个数学研究项目的目标是研究复变类(更一般地，Cauchy-Riemann(CR)结构)中一般实子流形的几何、解析和代数方面及其映射。艾本菲尔特将考虑有关CR图的存在、唯一性和规律性的问题，以及与本研究有关的相关问题。他将考虑从Levi非退化超曲面到高维非退化超二次曲面的CR映射的背景。这一研究将加深我们对超二次曲面CR子流形结构的理解，它构成了Levi非退化超曲面理论中的平坦模型。这项工作还应该深入了解这种超曲面的局部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，如非退化和刚性的概念。艾本费尔特还将继续他的无限类型子流形之间的CR映射的研究，通过研究定义到喷丛上的奇异Pfaffian系统的CR映射的系统的延拓。他将特别着重于理解最近发现的现象，即对于无限型超曲面，双全纯、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喷流决定的猜想。这些研究有望对无穷型流形之间的CR映射的性质提供新的启示，并有助于更好地理解在此背景下产生的Pfaffian系统。埃本费尔特还将继续他在复2-空间中无限类型超曲面的范式方面的工作。最后，艾本费尔特将研究Kähler流形上单位圆丛上的CR不变量，特别是三维情形下的Cartan脐张量以及Bergman和Szegökernels展开时产生的那些不变量。他打算发展新的方法来检测一般三维CR流形上的脐点，并解决关于嵌入在复2-空间中的紧致CR流形上脐点的存在的公开问题。

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.