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

摘要：项目负责人：John W. Morgan, Mikhail G. Khovanov,Walter D. Neumann, Peter S. ozsvath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、表示理论、全纯曲线理论、分析和代数几何。这种技术的融合带来了爆炸性的结果，也带来了新的研究途径。这方面的例子包括佩雷尔曼最近用几何分析方法证明庞加莱猜想；Donaldson对光滑四流形的构造不变量的规范理论方法，以及随后Seiberg-Witten方程的发现；全纯曲线定义的三流形和四流形不变量结的量子多项式不变量，以及最近的分类。该项目旨在培养这些令人兴奋和充满活力的学科的下一代研究人员，包括研究生和博士后；并广泛传播这些领域的最新发现。该提案的目的是激发哥伦比亚大学在低维拓扑和几何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率先开展的。该计划的目标是：(i)增加在这个令人兴奋和活跃的数学领域攻读研究生的本科生数量，并确保他们的培训为研究生学习做好准备；(2)拓宽和加强哥伦比亚大学研究生的研究背景，使他们能够更好地发挥自己的研究潜力，并为这一快速变化的学科做出有意义的贡献；（三）培养博士后在本领域进行更独立的研究；（iv）促进哥伦比亚大学研究小组各阶层之间的交流，并在论述技巧、教学和研究方面对相关学生和博士后进行培训；(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。

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.