State-Sum Methods in Low-Dimensional Topology and Quantum Gravity - Categorical Underpinnings and Applications

小行星9971510 本计画致力于低维拓扑学与量子重力学中与态和技术相关的多项相互关联且互补的研究。 其中一些涉及状态和技术的范畴基础：对TQFT的代数结构的研究，对具有结构的范畴的变形理论的研究，以及"状态积分“的范畴基础，即，测量熟悉的状态和的理论类似物。 其他的工作则是将这些技术应用于拓扑学和量子引力：扩展了欧几里得量子引力的Barrett-Crane态和公式的Minkowski签名，研究和构造了与Barrett-Crane和Crane-Yetter态和及其推广有关的拓扑不变量，研究了范畴形变理论与Vassiliev结和链不变量理论之间的关系。 这个项目旨在研究三维和四维空间的结构，包括拓扑学的抽象空间，以及引力量子理论的某些拟议模型中的物理时空结构。 量子化引力的问题可能是物理学中最深的未解决的理论问题。 它指出了我们对宇宙理解中最基本的未知数，比如宇宙的起源和黑洞物理学。 用传统的方法，它的明显的不可处理性表明，一种全新的数学类型可能是必要的，以解决这个问题。 量子态空间上算子的结构，描述空间或简单片段的结点和链接的集合的代数结构，以及构造具有适当不变性性质的状态和所需的代数数据，以研究拓扑结构和构造量子引力的“自旋泡沫”模型，都可以很方便地用Monoidal范畴理论的一般化和扩展来描述，Monoidal范畴理论是由MacLane在20世纪60年代初首先提出的。这些事实，再加上最近在上述相关问题上的进展，使我们希望monoidal categories理论可能是缺失的工具。 因此，大部分工作使用范畴论，而不是作为数学基础的集合论的替代品，而是作为研究从我们生活的空间中抽象出来的空间的适当代数。

9971510Yetter This project pursues a number of interrelated and complementarylines of research related to state-sum techniques in low-dimensionaltopology and quantum gravity. Some of these involve the categoricalunderpinnings of state-sum techniques: the investigation of the algebraicstructure of TQFT's, of the deformation theory of categories withstructure, and the categorical underpinnings of ``state-integrals,'' i.e.,measure theoretic analogues of the familiar state-sums. Others aim atapplications of the techniques to topology and quantum gravity: theextension to the Minkowskian signature of the Barrett-Crane state-sumformulation of Euclidean quantum gravity, the investigation andconstruction of topological invariants related to the Barrett-Crane andCrane-Yetter state-sums and their generalizations, and the investigationof the relations between categorical deformation theory and the theoryof Vassiliev knot and link invariants. This project seeks to investigate the structure of 3- and4-dimensional spaces, both the abstract spaces of topology, and thestructure of physical space-times in certain proposed models for aquantum theory of gravity. The problem of quantizing gravity is probablythe deepest unsolved theoretical problem in physics. It points to themost fundamental unknowns in our understanding of the universe, such asthe origin of the universe and black hole physics. Its apparentintractability by using traditional methods suggests that a radically newtype of mathematics may be necessary to solve the problem. The structureof operators on spaces of quantum states, the algebraic structuresdescribing the assembly of spaces or knots and links from simple pieces,and the algebraic data needed to construct state-sums with suitableinvariance properties to investigate both topological structures and toconstruct ``spin-foam'' models of quantum gravity, can all conveniently bedescribed in terms of generalizations and extensions of the theory ofmonoidal categories first introduced by MacLane in the early 1960's.These facts, together with recent progress with the interrelated problemsmentioned above, lead to the hope that the theory of monoidal categoriesmay be the missing tool. Thus, much of the work uses category theory,not as a substitute for set theory as a foundations of mathematics, butas the appropriate sort of algebra for the investigation of spacesabstracted from the one in which we live.***