EMSW21-RTG: Geometry, Topology, and Operator Algebras

奖项：dms -0838703首席研究员：robon C. Kirby， Peter teichner该奖项由2009年美国复苏与再投资法案（公法111-5）资助。主要研究人员（Robion Kirby和Peter Teichner）与其他高级人员（教授sian 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，博士后和高级教师组成的研究培训小组，研究几何，拓扑和算子代数领域。目标是通过在上述领域建立一个强大的、连贯的跨学科小组，改善本科生进入数学研究生学习的招聘，以及研究生和博士后的培养。这些领域是一致的，因为它们的中心问题和方法都来自数学物理，比如计算3维和4维流形的Seiberg-Witten和Heegaard-Floer不变量的伪全纯曲线的计数，或者使用里奇流来理解光滑流形，或者使用量子场论的数学概念来描述有趣的代数拓扑，比如椭圆上同调。今天，年轻人进入数学研究领域的一个主要困难是这门学科越来越多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方面的效果越来越差。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。这些将包括学年期间由研究生在高级教师的监督下领导的本科生研究研讨会，本科生和研究生夏季研讨会，初级研究生研讨会，热点研讨会，以及每周一次的联合研究研讨会，每次研究讲座之前都有背景讲座。几何学、拓扑学和算子代数的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师资队伍，并且在成功培养研究生和本科生方面有着悠久的历史。然而，这些学科变得越来越跨学科，要求研究生比过去了解更多，并且需要高级数学家更多的指导。为了提高对研究生和博士后的培养，他们也应该提高他们的研究质量。跨群互动将促进我们打算进行的一些跨学科研究项目，包括：椭圆上同调与量子场理论，k理论与拓扑场理论，椭圆上同调的半无限方面和四维场理论。

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.