Geometry and Physics

几何与物理

• 批准号: 0072675
• 负责人: Daniel Freed
• 金额: $ 62.28万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072675&HistoricalAwards=false
• 关键词: Geometry; Physics

AbstractAward: DMS-0072675Principal Investigator: Daniel S. FreedThe geometry group at the University of Texas proposes to carryout a variety of research projects, most of which are related tophysics. Dan Freed's current research focuses on questions ingeometry and topology arising from string theory. These includethe construction of determinant line bundles on manifolds withboundary and the understanding of anomalies of actions whicharise in quantizing lagrangian field theories. Karen Uhlenbeck'scurrent research involves the geometric theory of integrablesystems, as well as a non-linear Schroedinger equation arising inmacroscopic theories of a ferro-magnetic continuum. These tworesearchers are starting a project aimed at understanding theappearance of integrable systems in conformal field theories.Bob Williams is applying his expertise on attractors to theconstruction of tiling spaces. Constantin Teleman is pursuingseveral projects in the cohomology of infinite dimensional LieAlgebras. He and his coworkers are making progress on theMacDonald conjectures, and are also giving geometricinterpretations in terms of the Hodge cohomology of flagvarieties of loop groups. He is also interested in homotopyequivalences between holomorphic and continuous mappingspaces. Postdoctoral members of the group are Nurit Krausz, whois working on direct computations for quantum field theory inMinkowski space, and Adrian Vajiac who uses equivariantlocalization techniques to study topological quantum fieldtheories.At this point in time, geometry is a rapidly developing area ofmathematics. While research in geometry, like most of puremathematics, consists of the construction and development ofabstract concepts, the origins and ultimate applications forthese constructions are invariably examples and applications inmore applied fields. The influence of theoretical physics ongeometry is strong. For example, large numbers of geometers arecurrently working on questions related to quantum groups, mirrorsymmetry and quantum cohomology. Our group attempts not to workon problems which have already been identified by mathematiciansas central, but in contrast we look at current ideas in physicsof all sorts and then find, clarify, and work on themathematically interesting questions more directly. Dan Freed'swork on string theory connects the physics ideas of quantizationwith the mathematical subject of algebraic topology. His jointproject with Karen Uhlenbeck on the appearance of integrablesystems in certain quantum field theories requires anunderstanding of field theory, integrable systems, and the veryimportant and basic ideas of symmetry. Constantin Teleman's workdeeply involves fundamental ideas of symmetry, as well as delvinginto the question of how closely very messy functions can beapproximated qualitatively by polynomial-like objects. Some ofthe geometric ideas come from other branches of physics, such asthe ferro-magnetic equations studied by Uhlenbeck. The tilingspaces of William's come from beautiful examples such as thoseconstructed by Roger Penrose. Our efforts bring new ideas andtechniques into mathematics, rather than concentrating onprojects which are already popular.

摘要：项目编号：dms -0072675项目负责人：Daniel S. freed德克萨斯大学几何小组提出开展多种研究项目，其中大部分与物理相关。Dan Freed目前的研究重点是由弦理论引起的几何和拓扑问题。这包括在有边界流形上的行列式线束的构造和对量子化拉格朗日场理论中产生的作用异常的理解。Karen Uhlenbeck目前的研究涉及可积系统的几何理论，以及在铁磁连续体的宏观理论中产生的非线性薛定谔方程。这两位研究人员正在开始一个旨在理解共形场理论中可积系统的外观的项目。鲍勃·威廉姆斯（Bob Williams）正在将他在吸引子方面的专业知识应用于瓷砖空间的构建。Constantin Teleman正在研究无限维liealgebra的上同调。他和他的同事们正在麦克唐纳猜想上取得进展，并且还根据环群的标志变异的Hodge上同调给出了几何解释。他也对全纯和连续映射空间之间的同伦等价感兴趣。该小组的博士后成员是Nurit Krausz，他在闵可夫斯基空间中研究量子场论的直接计算，Adrian Vajiac使用等变局域化技术研究拓扑量子场论。在这个时间点上，几何是一个迅速发展的数学领域。虽然几何研究，像大多数纯粹数学一样，是由抽象概念的构造和发展组成的，但这些构造的起源和最终应用总是在更多应用领域的例子和应用。理论物理对几何的影响很大。例如，大量的几何学家目前正在研究与量子群、镜像对称和量子上同调有关的问题。我们的小组试图不去研究那些已经被数学家们认定为中心的问题，相反，我们关注物理学中各种各样的最新观点，然后更直接地发现、澄清和研究那些在数学上有趣的问题。丹·弗里德在弦理论方面的工作将量子化的物理概念与代数拓扑的数学主题联系起来。他与凯伦·乌伦贝克（Karen Uhlenbeck）合作的项目是在某些量子场论中出现可积系统，这需要对场论、可积系统以及非常重要和基本的对称性概念有一定的了解。Constantin Teleman的工作深入地涉及到对称的基本思想，同时也深入研究了一个问题，即非常杂乱的函数可以用多项式样的物体在定性上近似到什么程度。一些几何概念来自物理学的其他分支，比如乌伦贝克研究的铁磁方程。威廉的瓷砖空间来自于美丽的例子，比如罗杰·彭罗斯的作品。我们的努力为数学带来了新的思想和技术，而不是集中在已经流行的项目上。