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

合作研究：低维结和流形的共循环不变量

• 批准号: 0301095
• 负责人: J. Scott Carter
• 金额: --
• 依托单位: University of South Alabama
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301095&HistoricalAwards=false
• 关键词: Collaborative; Research; Cocycle; Invariants; Low

DMS-0301095J. Scott CarterIn 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.

DMS-0301095J。Scott Carter在之前的工作中，主要研究者构建了一个高度非平凡的结一致性组过滤，由嵌入4球中的gropes的高度索引，并部分由冯诺依曼的连续维度检测。由此产生的分级组仍然是未知的，主要研究者提出了各种方法来揭示这个组的结构。相关问题的链接协调和嵌入问题的2-球到4-manifolds.In第二部分的项目，主要研究者是试图给一个几何定义的椭圆上同调的修改Segal的“椭圆对象”。这些都是由拓扑空间X参数化的共形场论，特别是对于X中的每个圆，它们都与希尔伯特空间相关联。在超过15年的时间里，西格尔的方法都不能转化为上同调理论，因为梅耶-维多里斯原理的失败。新的想法是应用融合双模冯诺依曼代数，开发的康纳斯，使共形场理论“本地在X”。当X中的相应圆被分解时，融合被用于分解希尔伯特空间。这样的局部理论应该满足上同调理论的所有公理。项目的两个部分都涉及从理论物理到数学的概念。从历史上看，匡威关系更常见，其中数学概念（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，令人惊讶的数学预言（只有在非常罕见的情况下才能证明）来自理论物理学（如量子引力或共形场论）的考虑。因此，数学研究的最终重要性是将这些考虑纳入到良好理解的理论体系中。在这个项目的第一部分，首席研究员建议通过最初由冯·诺依曼提出的用于量子力学研究的技术，继续他对四维流形（最相关的相对论）的成功研究。在第二部分中，主要研究者提出完善的概念，共形场理论，使它导致一个几何定义的“椭圆上同调”。这种上同调是一个非常成功的工具，在数学和拟议的改进有可能导致拓扑理解的所有共形场理论。