EMSW21-RTG: Geometry, Topology, and Operator Algebras

摘要奖：DMS-0838703主要研究者：Robion C.该奖项由2009年美国复苏和再投资法案资助。（公法111-5）主要调查人员（Robion Kirby和Peter Teichner），与其他高级人员合作（教授伊恩·阿戈尔，罗伯特·布莱恩特，迈克尔·哈钦斯，沃恩·琼斯，约翰·洛特，康斯坦丁·泰尔曼和艾伦·温斯坦），建议建立一个由本科生，研究生，博士后研究员和加州大学伯克利分校数学系在几何，拓扑和算子代数领域的高级教师。 其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，提高本科生进入数学研究生学习的招聘率，以及研究生和博士后研究员的培训。 这些领域是一致的，因为它们的中心问题和方法来自数学物理，例如计算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Ricci流来理解光滑流形，或者使用量子场论的基本概念来描述有趣的代数拓扑，比如椭圆余同调。如今，年轻人进入数学研究领域的一个主要困难是学科的跨学科性质和所需技术工具的范围越来越大。传统的研讨会形式在培训方面越来越不有效。 我们将与数学系现有的强有力的计划，创造一系列新的和重组的活动，从本科水平，以先进的研究。 这将包括本科生的研究coursaries在学年由研究生领导下的资深教师，本科生和研究生暑期讲习班，小学水平的研究生研讨会，热点研讨会，每周一次的联合研究研讨会，其中每个研究讲座之前的背景lecture.The领域的几何，拓扑和运营商代数涵盖在这个建议是中央数学和物理学。 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，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，这些学科已经发展得越来越多的跨学科，要求研究生比过去知道得更多，并且需要高级数学家的更多指导。 为了提高研究生和博士后的培养质量，他们还应该提高研究成果的质量。 跨群相互作用将促进我们打算开展的许多跨学科研究项目，包括：椭圆上同调和量子场论、K理论和拓扑场论、椭圆上同调的半无限方面以及四维场论。

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.