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.

摘要奖项：DMS-0072675 首席研究员：Daniel S. Freed 德克萨斯大学几何小组提议开展各种研究项目，其中大部分与物理学相关。 丹·弗里德目前的研究重点是弦理论引起的几何和拓扑问题。 其中包括在有边界的流形上构造行列式线束，以及理解量化拉格朗日场论中出现的作用异常。 Karen Uhlenbeck 目前的研究涉及可积系统的几何理论，以及铁磁连续体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启动一个项目，旨在了解共形场论中可积系统的外观。鲍勃·威廉姆斯正在将他在吸引子方面的专业知识应用于平铺空间的构造。 Constantin Teleman 正在研究无限维李代数上同调的几个项目。他和他的同事正在麦克唐纳猜想方面取得进展，并且还根据环群的旗形簇的霍奇上同调给出了几何解释。 他还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包括努里特·克劳斯（Nurit Krausz）和阿德里安·瓦吉亚克（Adrian Vajiac），努里特·克劳斯（Nurit Krausz）正在研究明可夫斯基空间中量子场论的直接计算，而阿德里安·瓦贾克（Adrian Vajiac）则使用等变局域化技术来研究拓扑量子场论。此时此刻，几何学是一个快速发展的数学领域。 虽然几何研究与大多数纯数学一样，包括抽象概念的构造和发展，但这些构造的起源和最终应用始终是更多应用领域的示例和应用。理论物理对几何学的影响很大。 例如，大量几何学家目前正在研究与量子群、镜像对称和量子上同调相关的问题。我们的小组试图不去研究数学家已经确定为核心的问题，但相反，我们研究当前各种物理学思想，然后更直接地发现、澄清和研究数学上有趣的问题。 丹·弗里德（Dan Freed）关于弦理论的工作将量子化的物理思想与代数拓扑的数学主题联系起来。 他与 Karen Uhlenbeck 的联合项目涉及某些量子场论中可积系统的出现，需要理解场论、可积系统以及对称性的非常重要和基本的思想。 康斯坦丁·泰勒曼（Constantin Teleman）的工作深入涉及对称性的基本思想，并深入研究了多项式对象可以在多大程度上定性地近似非常混乱的函数的问题。一些几何思想来自物理学的其他分支，例如乌伦贝克研究的铁磁方程。 威廉的瓷砖空间来自美丽的例子，例如罗杰·彭罗斯建造的那些。 我们的努力将新的思想和技术带入数学中，而不是专注于已经流行的项目。