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.

这个项目解决了一些存在于复杂几何、偏微分方程式和数学物理之间的基本问题。统一的主题是由偏微分方程解、它们的几何障碍和它们的奇点定义的复杂结构和模数。这些方法建立在偏微分方程理论的基础上，但核心部分将是来自复杂分析的新技术和新思想的发展。复杂结构和模最早出现在纯数学中，但此后它们揭示了自己处于从应用数学到统计和高能物理的广泛现象的中心。事实上，本项目中的一些问题直接源于关于临界现象和超对称性的问题。项目任何部分的进展都应该产生广泛的影响，无论是在数学领域还是在相关领域。这位首席研究员过去培养了源源不断的本科生、研究生和博士后研究人员，该项目将是他继续朝着这个方向努力的关键。