Geometry and Physics

摘要奖：DMS-0072675主要研究人员：丹尼尔·S·弗里德德克萨斯大学几何学组提议开展各种研究项目，其中大部分与拓扑物理有关。丹·弗里德目前的研究集中在弦理论引起的计量学和拓扑学问题上。这些内容包括在有边界的流形上构造行列式线丛，以及理解在量子化拉格朗日场理论中所需的反常作用量。Karen Uhlenbeck目前的研究涉及可积系统的几何理论，以及铁磁连续统宏观理论中出现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出现。鲍勃·威廉姆斯正在将他在吸引子方面的专业知识应用于瓦片空间的构造。Constantin Teleman在无限维Lie代数的上同调中追求多个项目。他和他的同事们在麦克唐纳猜想方面取得了进展，并根据循环群的旗舰变种的Hodge上同调给出了几何解释。他还对全纯映射空间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在Minkowski空间从事量子场论直接计算的Nurit Krausz和使用等变局域化技术研究拓扑量子场理论的禤浩焯·瓦贾克。在这个时间点上，几何是一个迅速发展的数学领域。虽然几何学的研究像大多数纯数学一样，包括抽象概念的构建和发展，但这些结构的起源和最终应用总是例子和在更多应用领域的应用。理论物理对几何学的影响是很大的。例如，大量的几何学家目前正在研究与量子群、镜像对称性和量子上同调有关的问题。我们小组试图不研究那些已经被数学家确定为中心的问题，但与之相反，我们查看各种物理中的当前想法，然后更直接地发现、澄清和研究这些有趣的数学问题。丹·弗里德在弦理论方面的工作将量子化的物理思想与代数拓扑学的数学主题联系起来。他与凯伦·乌伦贝克就某些量子场论中可积系统的出现进行的联合项目需要理解场论、可积系统以及非常重要和基本的对称性概念。康斯坦丁·特尔曼的工作深入地涉及了对称性的基本概念，以及深入研究了非常杂乱的函数可以多大程度上接近于类多项式对象的问题。一些几何概念来自物理学的其他分支，例如乌伦贝克研究的铁磁方程。威廉的瓦片空间来自美丽的例子，比如罗杰·彭罗斯建造的那些。我们的努力为数学带来了新的思想和技术，而不是专注于已经很受欢迎的项目。

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.