Controlled Topology and Topological Field Theory

9705168奎恩这个项目有两个不同的组成部分，受控拓扑学和拓扑场论。受控拓扑学的目标是更好地理解由于回撤的形成受阻而导致的近似纤颤不能形成丛理论的原因。Hughes和其他人精确地解释了近似纤颤是同伦分层集的正常结构的意义。这使得回调问题成为这些集合研究的核心。通过休斯、威廉姆斯、泰勒、温伯格和其他人的工作，人们已经知道了很多东西，但更令人满意的版本可能是可能的。另一个主题涉及定义在2-复形、3-流形和光滑4-流形上的拓扑场论中的机器计算。2-复数情形已经完全实现，来自李代数的mod p表示的场论的探索进展顺利，p可达13。3-流形所需的量子群的表示理论已经实现，但场论的探索还有待于对2-复数情形的更全面的理解。2-复形和3-流形都被认为是该项目的主要目标--4-流形上的场论--的测试用例。然而，这还需要几年的时间。一些拓扑学方面的工作也在进行，场论可能对这些问题有所启发：2-复形的“Andrews-Curtis”猜想，4-流形上的把手结构，以及4-流形的拓扑同构。第一个主题的背景是对分层集合的研究。自然界中出现的大多数几何对象都是分层的：由流形的层组成。最大的挑战是了解这些地层是如何组合在一起的。对于大多数类型的分层集(光滑的、分段线性的、解析的)，使用丛理论来描述这种拟合。它很复杂，很难使用，因为对象很复杂，但很有效，很容易理解。顶层分层物体的描述花费了更长的时间，因为事实证明，拼合在一起不能用集束理论来给出。这方面已经进行了广泛的研究，但我们还没有一个完全令人满意的理解。项目第二部分中的场论也被称为“拓扑量子场论”。它们是专门针对低维的、相当独特的理论。正在研究的那些是代数的(与解析相反)，并且使用群或李代数或它们的变形的表示来构造。这些，特别是3-歧式版本，在大约10年前变得流行起来。当时，我们似乎既不了解李代数的变形，也不了解3-流形，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。该项目目前已进入第八个年头，将数值计算作为第一个目标。这一切进展顺利，但原始产出并不是很能说明问题。目前的主要活动是寻找使用数字数据的方法，以获得对理论的全球定性理解。***

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. ***