Geometry and Physics

摘要奖：DMS-0072675主要研究者：丹尼尔S.得克萨斯大学的几何学小组提议进行各种各样的研究项目，其中大部分与物理学有关。 丹·弗里德目前的研究主要集中在弦理论中的几何学和拓扑学问题上。 这包括在有边界流形上的行列式线丛的构造和对量子化拉格朗日场论中的作用量反常的理解。 卡伦·乌伦贝克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的非线性薛定谔方程。这两个研究者正在开始一个旨在理解共形场论中可积系统的出现的项目。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正在研究无穷维李代数的上同调。他和他的同事们在MacDonald图上取得了进展，并且还用环群的旗簇的Hodge上同调给出了几何解释。 他还感兴趣的同伦之间的全纯和连续mappingspaces。该小组的博士后成员有Nurit Krausz，他正在研究Minkowski空间中量子场论的直接计算，以及Adrian Vajiac，他使用等变局部化技术研究拓扑量子场论。在这个时候，几何是一个快速发展的数学领域。 虽然几何学的研究，像大多数纯数学一样，由抽象概念的构建和发展组成，但这些构建的起源和最终应用总是在更多应用领域中的例子和应用。理论物理学对几何学的影响很大。 例如，大量的几何学家正在研究与量子群、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确定为中心的问题，相反，我们研究各种物理学中的现有思想，然后更直接地发现、澄清和研究数学上有趣的问题。 丹·弗里德在弦理论上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。 他与卡伦乌伦贝克的联合项目的外观积分系统在某些量子场论需要理解场论，可积系统，和非常重要的基本思想的对称性。 Constantin Teleman的作品深入涉及对称性的基本思想，以及深入研究如何密切非常混乱的函数可以近似定性的多项式样对象的问题。一些几何思想来自物理学的其他分支，如乌伦贝克研究的铁磁方程。 威廉的拼贴空间来自于罗杰·彭罗斯的那些美丽的例子。 我们的努力将新的思想和技术带入数学，而不是集中在已经流行的项目上。

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.