EMSW21-RTG: New Techniques in Low-Dimensional Topology and Geometry

EMSW21-RTG：低维拓扑和几何新技术

• 批准号: 0739392
• 负责人: Mikhail Khovanov
• 金额: $ 249.93万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0739392&HistoricalAwards=false
• 关键词: EMSW21; RTG; New; Techniques; Low

AbstractAward: DMS 0739392Principal Investigator: John W. Morgan, Mikhail G. Khovanov,Walter D. Neumann, Peter S. OzsvathLow-dimensional topology is undergoing a revolution thanks to aninfusion of new techniques from other areas, includingmathematical physics, representation theory, holomorphic curvetheory, analysis, and algebraic geometry. This fusion oftechniques has led to an explosion of results and also newavenues of research. Examples of this include Perelman's recentproof of the Poincare conjecture using methods from geometricanalysis; Donaldson's gauge theory approach to constructinvariants for smooth four-manifolds, and the subsequentdiscovery of the Seiberg-Witten equations; invariants of threeand four-manifolds defined via holomorphic curves; the quantumpolynomial invariants for knots and, more recently, theircategorifications. This project aims to train the nextgeneration of researchers in these exciting and vibrant subjects,including graduate students and postdocs; and also to widelydisseminate the latest discoveries in these fields.The aim of this proposal is to stimulate the research trainingenvironment at Columbia University in the new developments inlow-dimensional topology and geometry. Many of these developmentshave been pioneered by faculty at Columbia. The objectives ofthis program are: (i) to increase the number of undergraduateswho will pursue graduate studies in this exciting and active areaof mathematics, and to ensure that their training equips themwell for graduate studies; (ii) to broaden and strengthen theresearch background of the graduate students at Columbia, so thatthey are better equipped to fulfill their research potential, andcontribute meaningfully to this rapidly-changing subject; (iii)to prepare the postdoctoral associates for more independentresearch in this area; (iv) to foster communication among allstrata in the research group at Columbia, and in fact to trainthe associated students and postdocs in expository skills,teaching, and research; and (v) to foster the dissemination ofknowledge in this field from Columbia University to the widermathematical community.

摘要奖：DMS 0739392主要研究者：John W.作者声明：Michael G.作者：Walter D.作者：Peter S.低维拓扑学正在经历一场革命，这要归功于来自其他领域的新技术的注入，包括数学物理，表示论，全纯曲线理论，分析和代数几何。这种技术的融合导致了结果的爆炸，也带来了新的研究场所。 这方面的例子包括佩雷尔曼最近的证明庞加莱猜想使用的方法从geometricanalysis;唐纳森的规范理论的方法来构建光滑的四流形的variants，和随后发现的塞伯格-威滕方程;不变量的threeand四流形定义通过全纯曲线; quantumpolynomial不变量的结，最近，theircalculations。 该项目旨在培养下一代研究人员，包括研究生和博士后，并广泛传播在这些领域的最新发现，该建议的目的是激发哥伦比亚大学在低维拓扑学和几何学新发展方面的研究培训环境。其中许多发展都是由哥伦比亚大学的教师开创的。该计划的目标是：（i）增加本科生谁将继续研究生课程在这一令人兴奋的和活跃的数学领域，并确保他们的培训装备他们以及研究生课程;（ii）扩大和加强研究生在哥伦比亚的研究背景，使他们更好地装备，以实现他们的研究潜力，并有意义地贡献这一迅速变化的主题;（iii）为博士后研究员在这一领域进行更独立的研究做好准备;（iv）促进哥伦比亚研究小组各阶层之间的交流，实际上是在短期技能、教学和研究方面培训相关的学生和博士后;（v）促进哥伦比亚大学向更广泛的数学界传播这一领域的知识。