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

9971510然而，本项目追求与低维拓扑和量子引力的状态和技术相关的一些相互关联和互补的研究方向。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，具有结构的范畴的变形理论，以及“状态积分”的范畴基础，即熟悉的状态和的测量理论类似物。其他研究的目的是将这些技术应用于拓扑和量子引力：将欧几里得量子引力的Barrett-Crane状态求和公式推广到闵可夫斯基签名，研究和构造与Barrett-Crane和crane - yetter状态和及其推广相关的拓扑不变量，以及研究分类变形理论与Vassiliev结和连接不变量理论之间的关系。该项目旨在研究三维和四维空间的结构，既包括拓扑的抽象空间，也包括某些量子引力理论模型中的物理时空结构。引力的量子化问题可能是物理学中最深奥的未解理论问题。它指出了我们对宇宙的理解中最基本的未知数，比如宇宙的起源和黑洞物理学。用传统方法解决这个问题的明显性表明，可能需要一种全新的数学来解决这个问题。量子态空间上的算子的结构，描述由简单块组成的空间或结点和链接的代数结构，以及构造具有适当不变性的状态和以研究拓扑结构和构造量子引力的“自旋泡沫”模型所需的代数数据，都可以用MacLane在20世纪60年代初首次引入的一元范畴理论的推广和扩展来方便地描述。这些事实，加上最近在上述相关问题上取得的进展，使人们希望一元范畴理论可能是缺失的工具。因此，范畴论并不是作为集合论作为数学基础的替代品，而是作为一种适当的代数来研究从我们生活的空间中抽象出来的空间

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.***