EMSW21-RTG: Geometry, Topology, and Operator Algebras

摘要奖：DMS-0838703主要研究人员：Robion C.Kirby，Peter Teichner该奖项由2009年《美国复苏和再投资法案》(Public Law 111-5)资助。主要研究人员(Robion Kirby和Peter Teichner)与其他高级人员(教授Ian Agol、Robert Bryant、Michael Hutchings、Vaughan Jones、JohnLott、Constantin Teleman和Alan Weinstein)合作，提议在加州大学伯克利分校数学系建立一个由本科生、研究生、博士后研究员和高级教员组成的几何、拓扑和算子代数领域的研究培训小组。其目标是通过在上述领域建立一个强有力的、协调一致的跨学科小组，改进数学研究生研究的本科生招生工作以及研究生和博士后研究员的培训工作。这些领域是连贯的，因为它们的中心问题和方法来自数学物理，例如计算伪全纯曲线的计数来计算三维和四维流形的Seiberg-Witten和Heegaard-Floer不变量，或者使用Ricci流来理解光滑流形，或者使用量子场论的数学概念来描述有趣的代数拓扑，如椭圆上同调。今天年轻人进入数学研究的一个主要困难是学科的跨学科性质和所需的技术工具的范围。传统的研讨会形式在培训中变得不那么有效。我们将与数学系现有的强大项目合作，创建一系列新的和重组的活动，从本科水平到高级研究。这些课程将包括本学年在高级教师指导下由研究生主持的本科生研究研讨会、本科生和研究生暑期工作坊、初级研究生研讨会、热门话题研讨会和每周联合研究研讨会，在每一次研究演讲之前都有一堂背景讲座。这项建议涵盖的几何、拓扑和算子代数领域是数学和数学物理的核心。加州大学伯克利分校在这一领域拥有一支杰出的教师队伍，在成功培养研究生和本科生方面有着悠久的历史。然而，这些学科越来越多地跨学科发展，要求研究生比过去了解更多的知识，需要高级数学家提供更多的指导。为了改善对研究生和博士后的培训，他们还应该提高他们所产生的研究质量。跨基团的相互作用将促进我们打算从事的一些跨学科研究项目，包括：椭圆上同调和量子场理论，K-理论和拓扑场理论，椭圆上同调的半无限方面，以及四维场理论。

AbstractAward: DMS-0838703Principal Investigator: Robion C. Kirby, Peter TeichnerThis award is funded under the American Recovery and ReinvestmentAct of 2009 (Public Law 111-5).The principal investigators (Robion Kirby and Peter Teichner),with the cooperation of the other senior personnel (ProfessorsIan Agol, Robert Bryant, Michael Hutchings, Vaughan Jones, JohnLott, Constantin Teleman, and Alan Weinstein), propose toestablish a research training group of undergraduates, graduatestudents, postdoctoral fellows, and senior faculty at the UCBerkeley mathematics department in the areas of geometry,topology and operator algebras. The objectives are to improvethe recruitment of undergraduates into graduate studies inmathematics, and the training of graduate students andpostdoctoral fellows, by establishing a strong, coherentinterdisciplinary group in the above areas. These areas arecoherent in that they have at their center problems and methodsfrom mathematical physics, such as the count of pseudoholomorphiccurves to calculate Seiberg-Witten and Heegaard-Floer invariantsof 3- and 4-dimensional manifolds, or the use of Ricci flow forunderstanding smooth manifolds, or the use of mathematicalnotions of quantum field theories to describe interestingalgebraic topology, like elliptic cohomology.A major difficulty for young people entering research inmathematics today is the increasingly interdisciplinary nature ofthe subject and the range of technical tools required.Traditional seminar formats are becoming less effective attraining. We will work with the existing strong programs of themathematics department to create a series of new and restructuredactivities, ranging from the undergraduate level to advancedresearch. These will include undergraduate research seminarsduring the school year led by graduate students under thesupervision of senior faculty, undergraduate and graduate summerworkshops, elementary level graduate student seminars, a hottopic seminar, and a weekly joint research seminar in whicheach research talk is preceded by a background lecture.The areas of geometry, topology and operator algebras covered inthis proposal are central to mathematics and mathematicalphysics. UC Berkeley has an exceptional group of faculty in thisarea, and a long history of successfully training graduate andundergraduate students. However, these subjects have grown moreand more interdisciplinary requiring graduate students to knowmuch more than in the past, and more guidance is required ofsenior mathematicians. To improve the training of graduatestudents and postdocs, they should also improve the quality ofresearch that they produce. The cross-group interaction willfacilitate a number of interdisciplinary research projects thatwe intend to pursue, including: Elliptic cohomology and quantumfield theory, K-theory and topological field theories,Semi-infinite aspect of elliptic cohomology, and Four-dimensionalfield theories.