Controlled Topology and Topological Field Theory

受控拓扑和拓扑场论

• 批准号: 9705168
• 负责人: Frank Quinn
• 金额: $ 13.3万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9705168&HistoricalAwards=false
• 关键词: Controlled; Topology; Topological; Field; Theory

9705168 Quinn This project has two distinct components, in controlled topology and topological field theory. The controlled topology objective is to understand better the failure of approximate fibrations to form a bundle theory because the formation of pullbacks is obstructed. Hughes and others have made precise the sense in which approximate fibrations are the normal structure in homotopically stratified sets. This makes the pullback problem central in the study of these sets. Much is already known through the work of Hughes, Williams, Taylor, Weinberger and others, but a more satisfactory version may be possible. The other topic concerns machine computation in topological field theories defined on 2-complexes, 3-manifolds, and smooth 4-manifolds. The 2-complex case is completely implemented, and explorations are well along for field theories coming from mod p representations of Lie algebras, for p up to 13. The representation theory of quantum groups needed for 3-manifolds has been implemented, but exploration of the field theories waits on a more complete understanding of the 2-complex case. Both 2-complexes and 3-manifolds are considered test cases for the primary goal of the project, field theories on 4-manifolds. This, however, is still some years off. Some work is also being done on topological questions on which field theories might shed light: the "Andrews-Curtis" conjecture for 2-complexes, handlebody structures on 4-manifolds, and topological isotopy of 4-manifolds. The context for the first topic is the study of stratified sets. Most of the geometric objects occurring in nature are stratified: built up of layers that are manifolds. The biggest challenge is to understand how the strata fit together. For most types of stratified sets (smooth, piecewise-linear, analytic) this fitting-together is described using a bundle theory. It is complex and hard to use because the objects are complicated, but is effective and well-understood. Top ological stratified objects took longer to describe, because it turns out the fitting-together cannot be given in terms of a bundle theory. This has been extensively studied, but we do not yet have a fully satisfactory understanding. The field theories in the second part of the project are also referred to as "topological quantum field theories." They are rather idiosyncratic theories specialized to low dimensions. The ones under study are algebraic (as opposed to analytic) and are constructed using representations of groups or Lie algebras, or their deformations. These, particularly the 3-manifold versions, became popular about a decade ago. At the time, we seemed not to know enough either about deformations of Lie algebras or about 3-manifolds to get significant new information from one about the other. A particular obstacle was the inability to compute "random" invariants. This project, now in its eighth year, has numerical computation as a first goal. This is going well, but the raw output is not very revealing. The principal activity at present is finding ways to use numerical data to obtain a global qualitative understanding of a theory. ***

小行星9705168 这个项目有两个不同的组成部分，在控制拓扑学和拓扑场论。 控制拓扑的目标是更好地理解失败的近似纤维化，形成一个捆绑的理论，因为形成的拉回是阻碍。 Hughes和其他人精确地指出，近似纤维化是同伦分层集合的正规结构。 这使得拉回问题在这些集合的研究中处于中心地位。 通过休斯、威廉姆斯、泰勒、温伯格和其他人的工作，我们已经知道了很多，但可能会有一个更令人满意的版本。 另一个主题是关于定义在2-复形、3-流形和光滑4-流形上的拓扑场论的机器计算。 2-复杂的情况下是完全实现的，探索是沿着为场论来自模p表示的李代数，p高达13。 三维流形所需的量子群的表示理论已经实现，但对场论的探索还有待于对二维复形的更完整的理解。 2-复形和3-流形都被认为是该项目的主要目标的测试用例，4-流形上的场论。 然而，这还需要几年的时间。 一些工作也正在做的拓扑问题上的领域理论可能揭示：“安德鲁斯-柯蒂斯”猜想2复合物，体结构的4流形，拓扑合痕的4流形。 第一个主题的背景是分层集的研究。 自然界中出现的大多数几何对象都是分层的：由流形层组成。 最大的挑战是了解地层是如何组合在一起的。 对于大多数类型的分层集（光滑，分段线性，解析），这种拟合在一起描述使用束理论。 由于对象很复杂，因此它很复杂且难以使用，但它很有效且易于理解。 拓扑分层的对象需要更长的时间来描述，因为它证明了在捆绑理论中不能给出拟合。 这一点已得到广泛研究，但我们尚未有一个完全令人满意的理解。 该项目第二部分的场论也被称为“拓扑量子场论”。“它们是专门针对低维的相当特殊的理论。 正在研究的那些是代数的（与分析相反），并且使用群或李代数或其变形的表示来构造。 这些，特别是三歧管版本，大约十年前开始流行。 当时，我们似乎对李代数的变形或三维流形都没有足够的了解，无法从其中一个获得关于另一个的重要新信息。 一个特别的障碍是无法计算“随机”不变量。 这个项目，现在在其第八个年头，数值计算作为第一个目标。 这一切进展顺利，但原始产出并不十分明显。 目前的主要活动是寻找使用数值数据的方法来获得对理论的全面定性理解。 ***