Problems in Complex Analysis and Complex Geometry

复杂分析和复杂几何问题

• 批准号: 1266033
• 负责人: Duong Phong
• 金额: $ 77.6万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1266033&HistoricalAwards=false
• 关键词: Problems; Complex; Analysis; Geometry

This project addresses some fundamental problems that lie at the interface of complex geometry, partial differential equations, and mathematical physics. The unifying theme is complex structures and moduli defined by solutions of partial differential equations, their geometric obstructions, and their singularities. The methods build on the theory of partial differential equations, but the central component will be the development of new techniques and ideas from complex analysis.Complex structures and moduli first appeared in pure mathematics, but they have since revealed themselves to be at the center of a wide range of phenomena, from applied mathematics to both statistical and high-energy physics. In fact, some of the problems in the present project arise directly from questions about critical phenomena and supersymmetry. Progress in any part of the project should have wide repercussions, both in mathematics and in related fields. The principal investigator has trained a steady stream of undergraduates, graduate students, and postdoctoral researchers in the past, and the project will be crucial in allowing him to continue his efforts in this direction.

该项目解决了一些基本问题，这些问题位于复杂几何，部分微分方程和数学物理学的界面上。统一的主题是由部分微分方程的解决方案，它们的几何障碍物及其奇异性定义的复杂结构和模量。这些方法基于部分微分方程的理论，但是中心部分将是复杂分析中的新技术和思想的发展。实际上，本项目中的某些问题直接来自有关关键现象和超对称性的问题。在数学和相关领域，项目的任何部分的进步都应产生广泛的影响。过去，首席研究人员曾培训过稳定的本科生，研究生和博士后研究人员，该项目对于允许他继续朝这个方向努力至关重要。