Mappings in Several Complex Variables and CR geometry

彼得·埃本费尔特的这个数学研究项目的目标是研究复变集(更一般地，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。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。Ebenfit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。这一研究将加深我们对超二次曲面CR子流形结构的理解，超二次曲面本身构成了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类超曲面的局部CR几何(原则上完全编码在CR曲率张量中)如何影响其CR映射的性质，例如，非退化和刚性的各种概念。即使在源流形和目标流形都是超二次曲面的特殊情况下(例如Gap猜想)，也存在非常有趣和非平凡的问题。埃本菲尔特将继续他对无限(交换子)型一般子流形之间的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。PI将特别关注在此上下文中关于局部自同构的有限喷射确定的一个猜想。这一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。埃本费尔特将继续他关于复2-空间中无限型超曲面的正规形的工作。在这个方向上的结果将产生关于有限喷流确定问题的结果，并且可能产生在这种情况下形式映射的收敛问题的结果。艾本费尔特还将研究更一般的CR流形之间的CR映射：一个令人感兴趣的主题是横截性，他将努力将最近的横截性结果(与Duong和Baouendi-Rothschild一起)改进到更高维的CR映射。本文中的横截性的当前条件涉及Levi形式的特征值。艾本费尔特认为，实际上涉及到CR结构的“更深层次”不变量，就像等维情形一样。这项研究可能需要开发大量新的方法，这反过来将有利于CR映射到更高维空间的理论。复流形中实子流形的研究是复分析以及数学和物理的其他领域的核心。在这个数学研究项目中，使用并进一步发展了来自广泛领域的工具，如实数和复数分析、偏微分方程组和代数几何。该项目所进行的研究将有助于邻近数学领域的研究以及理论物理领域的研究。所开发的方法和技术将在数学的其他领域有用，也可能在物理学(例如弦论)和工程学(例如控制论；系统工程)中有用。艾本费尔特预计，该项目将为研究生和博士后提供有趣的研究主题。由该项目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。

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 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. Ebenfelt will consider 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 themselves constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The work should also provide insight into how the local CR geometry of such hypersurfaces (in principle completely encoded in the CR curvature tensor) affects properties, such as e.g. various notions of nondegeneracy and rigidity, of their CR maps. There are highly interesting and nontrivial problems even in the special case where both the source and target manifolds are hyperquadrics (e.g. the Gap Conjecture). Ebenfelt will continue his study of CR maps between generic submanifolds of infinite (commutator) type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. The PI will, in particular, focus on a conjecture in this context regarding finite jet determination of local automorphisms. The study is 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 continue his work on normal forms for infinite type hypersurfaces in complex 2-space.; results in this direction will yield results on the finite jet determination problem and, possibly, on the problem of convergence of formal mappings in this context. Ebenfelt will also study CR maps between more general CR manifolds: one topic of interest is that of transversality, and he will try to improve recent transversality results (joint with with Duong and Baouendi--Rothschild) for CR maps into a higher dimensional space. Current conditions for transversality in this context involve eigenvalues of the Levi form. Ebenfelt believes that "deeper" invariants of the CR structures are actually involved, as in the equidimensional case. This study will likely require development of substantially new methods, which in turn will benefit the theory of CR maps into higher dimensional spaces. The study of real submanifolds in complex manifolds is central to complex analysis and to other areas of mathematics and physics. In this mathematics research project, tools from a wide range of areas such as real and complex analysis, partial differential equations, and algebraic geometry are used and further developed. The investigations carried out in this project will benefit research in adjacent areas of mathematics as well as in areas of theoretical physics. The methods and techniques developed will be useful in other areas of mathematics, and likely also in physics (e.g., string theory) and engineering (e.g., control theory; systems engineering). Ebenfelt expects that the project will 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.