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

摘要奖：DMS 0739392主要研究员：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性增长，也带来了新的研究场所。这方面的例子包括Perelman最近用几何分析方法证明了Poincare猜想；Donaldson的规范理论方法来证明光滑四维流形的构造不变量，以及随后发现的Seiberg-Witten方程；通过全纯曲线定义的三维和四维流形的不变量；纽结的量子多项式不变量以及最近的它们的分类。该项目旨在培养下一代研究人员在这些令人兴奋和充满活力的学科，包括研究生和博士后；并广泛传播这些领域的最新发现。这项建议的目的是刺激哥伦比亚大学在低维拓扑和几何方面的新发展的研究培训环境。这些发展中的许多都是由哥伦比亚大学的教职员工开创的。该计划的目标是：(I)增加将在这一令人兴奋和活跃的数学领域攻读研究生课程的本科生人数，并确保他们的培训为研究生学习做好准备；(Ii)拓宽和加强哥伦比亚大学研究生的研究背景，使他们更好地准备好发挥他们的研究潜力，并为这一迅速变化的学科做出有意义的贡献；(Iii)为博士后助理做好准备，以便在该领域进行更多的独立研究；(Iv)促进哥伦比亚大学研究组各阶层之间的交流，事实上培训相关学生和博士后在暴露技能、教学和研究方面；以及(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.