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: 0301089
• 负责人: Masahiko Saito
• 金额: $ 12.7万
• 依托单位: University of South Florida
• 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 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301089&HistoricalAwards=false
• 关键词: Collaborative; Research; Cocycle; Invariants; Low

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.

DMS-0301089MASAHIKO SAITOIN先前的工作，首席研究人员构建了一个高度非平地的过滤，这是由嵌入4球中的螺纹高度索引的，并由冯·诺伊曼（Von Neumann）的连续尺寸部分检测到。由此产生的分级组仍然未知，主要研究者提出了各种方法来揭示该组的结构。这些方法也可以研究有关链接一致性和嵌入2次问题的问题的相关问题。西加尔的“椭圆对象”。这些是通过拓扑空间x参数参数的共形场理论，特别是x中的每个圆圈，它们与希尔伯特空间相关联。在15年以上的时间里，由于Mayer-Vietoris原则的失败，Segal的方法无法转变为同事理论。新想法是应用由Connes开发的von Neumann代数的双模型的融合，以使保形场理论“ X中的局部”。每当分解X中的相应圆圈时，融合用于分解Hilbert空间。然后，这种本地理论应该满足共同体理论的所有公理。项目的一部分将从理论物理学到数学的概念联系起来。从历史上看，相反的关系更为普遍，其中使用数学概念（例如Riemannian几何形状或功能分析）来解释物理理论（例如相对论或量子力学）。在过去的几十年中，令人惊讶的数学预测（仅在极少数情况下证明）在理论物理学（例如量子重力或保形场理论）中不考虑。因此，对于数学研究而言，将这些考虑因素纳入理解的主体是最终的重要性。在该项目的第一部分中，主要研究人员建议继续成功地研究他对4维流形的研究（相对论最相关）通过冯·诺伊曼（Von Neumann）最初提出的量子力学研究的技术。在第二部分中，主要研究者建议完善保形场理论的概念，以便导致“椭圆同学”的几何定义。这种同种学是一种在数学方面取得了巨大成功的工具，所提出的改进有可能导致对所有保形场理论的拓扑理解。