CAREER: Intrinsic and Extrinsic Conditions in Several Complex Variables

这个项目涉及复数分析，它结合了复数和微积分理论。在许多应用中，复杂分析是一种基本工具。特别是，它被用于物理学(例如：研究空气通过机翼的流动和光学中的色散关系)、工程(例如：信号处理和控制理论)和计算机科学(例如：图像处理和量子计算)。单变量的复杂分析是一个经典而又广为人知的数学课题，但当引入更多变量时，许多谜团依然存在。在这个项目中，PI将进一步加深对多变量复杂分析的理论理解。这个项目也有一个重要的教育组成部分：PI将为路易斯安那州立大学(LSU)的一年级本科生开发数据科学研究经验。该计划将为学员在STEM的职业生涯做好准备，巩固他们的基础数学知识，并给他们一个学习数据科学的机会。在这个项目的研究部分，PI将研究复欧氏空间中的有界域，并将内在几何条件(例如，Kaehler-Einstein度量的几何)与外部几何条件(例如，边界的CR几何)联系起来。本课题中的许多经典结果考虑了复欧氏空间中的有界域，假设边界是光滑的，并对边界的CR几何作了假设。对于具有非光滑边界的区域，PI将通过假设Kaehler-Einstein度规上的几何条件而不是边界上的条件来证明经典结果的新变体。使用这种方法，PI将能够研究通常超出标准分析方法范围的领域类别，并在旧问题上取得进展。这种方法的动机是几何群论的巨大成功，其中度量空间技术应用于群论不仅导致在旧问题上的进展，而且还带来了许多新的和有趣的例子。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns complex analysis, which combines complex numbers with the theory of calculus. Complex analysis is a fundamental tool in many applications. In particular, it is used in physics (for instance: studying the flow of air past an airfoil and dispersion relations in optics), engineering (for instance: signal processing and control theory), and computer science (for instance: image processing and quantum computation). Complex analysis of a single variable is a classical and well understood mathematical subject, but when additional variables are introduced many mysteries remain. In this project, the PI will further the theoretical understanding of complex analysis of several variables. This project also has a substantial educational component: The PI will develop a research experience in data science for first year undergraduate students at Louisiana State University (LSU). This program will prepare participants for a career in STEM, solidify their knowledge of basic mathematics, and give them an opportunity to learn about data science. In the research part of this project, the PI will study bounded domains in complex Euclidean space and relate intrinsic geometric conditions (e.g. the geometry of the Kaehler-Einstein metric) to extrinsic geometric conditions (e.g. the CR-geometry of the boundary). Many classical results in the subject consider bounded domains in complex Euclidean space, assume that the boundary is smooth, and make assumptions about the CR-geometry of the boundary. For domains with non-smooth boundary, the PI will prove new variants of classical results by assuming geometric conditions on the Kaehler-Einstein metric instead of conditions on the boundary. Using this approach, the PI will be able to study classes of domains which are typically outside the reach of the standard analytic methods and also make progress on old problems. This approach is motivated by the great success of geometric group theory where metric space techniques applied to group theory not only lead to progress on old problems, but also many new and interesting examples.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.