Collaborative Research: Cocycle Invariants of Low-Dimensional Knots and Manifolds

在之前的工作中，首席研究员构建了一个高度非平凡的结一致性群过滤，通过嵌入在4球中的葡萄的高度来索引，并通过冯·诺伊曼的连续维数来部分检测。最终的分级组仍然是未知的，首席研究员提出了各种方法来揭示这一组的结构。利用这些方法还可以研究2球在4流形中的链接一致性和嵌入问题。在项目的第二部分，主要研究者试图根据Segal的“椭圆对象”的修改给出椭圆上同的几何定义。这些是由拓扑空间X参数化的共形场论，特别是X中的每个圆都与希尔伯特空间相关联。在超过15年的时间里，由于Mayer-Vietoris原理的失败，Segal的方法无法转化为上同调理论。新的想法是应用由Connes发展的von Neumann代数的双模融合，使共形场论“局部于X”。当X中对应的圆被分解时，利用融合来分解希尔伯特空间。这样的局部理论应该满足上同调理论的所有公理。这个项目的两个部分都涉及到从理论物理到数学的概念。从历史上看，逆关系更常见，在数学概念（如黎曼几何或泛函分析）被用来解释物理理论（如相对论或量子力学）。在过去的几十年里，理论物理学（如量子引力或共形场论）中出现了令人惊讶的数学预测（只有在非常罕见的情况下才能证明）。因此，对数学研究来说，将这些考虑纳入人们很好理解的理论体系是极其重要的。在这个项目的第一部分，首席研究员建议通过冯·诺伊曼最初为研究量子力学而提出的技术，继续他对四维流形（与相对论最相关）的成功研究。在第二部分，主要研究者提出了完善共形场论的概念，使其导致“椭圆上同调”的几何定义。这个上同调在数学上是一个非常成功的工具，提出的改进有可能导致对所有共形场论的拓扑理解。

DMS-0301089Masahiko SaitoIn previous work, the principal investigator constructed a highly nontrivial filtration of the knot concordance group, indexed by the height of gropes embedded in the 4-ball, and partially detected by von Neumann's continuous dimension. The resulting graded group remains unknown, and the principal investigator proposes various approaches to uncover the structure of this group. Related questions about link concordance and embedding problems of 2-spheres into 4-manifolds can also be studied by these methods.In the second part of the project, the principal investigator is attempting to give a geometric definition of elliptic cohomology in terms of a modification of Segal's "elliptic objects". These are conformal field theories parametrized by a topological space X, in particular to each circle in X they associate a Hilbert space. For more than 15 years, Segal's approach could not be turned into a cohomology theory because of the failure of the Mayer-Vietoris principle. The new idea is to apply the fusion of bimodules of von Neumann algebras, developed by Connes, to make the conformal field theory "local in X". Fusion is used to decompose the Hilbert space whenever the corresponding circle in X is decomposed. Such a local theory should then satisfy all the axioms of a cohomology theory.Both parts of the project relate notions from theoretical physics to mathematics. Historically, the converse relation was more common, where a mathematical notion (like Riemannian geometry or functional analysis) was used to explain a physical theory (like relativity or quantum mechanics). In the last decades, surprising mathematical predictions (provable only in very rare cases) came out of considerations in theoretical physics (like quantum gravity or conformal field theory). It is thus of the ultimate importance for mathematical research to incorporate such considerations into the body of well understood theories.In the first part of this project, the principal investigator proposes to continue his successful study of 4-dimensional manifolds (most relevant in relativity) via techniques originally proposed by von Neumann for the study of quantum mechanics. In the second part, the principal investigator proposes to refine the notion of a conformal field theory so that it leads to a geometrical definition of "elliptic cohomology". This cohomology is an enormously successful tool in mathematics and the proposed refinement has the potential to lead to a topological understanding of all conformal field theories.