4-manifolds, von Neumann Algebras and Elliptic Cohomology

DMS-0305280 Peter Teichner在以前的工作中，主要研究人员构造了一个高度非平凡的纽结和谐群的滤子，以嵌入在4-球中的凹槽的高度为索引，并由冯·诺伊曼的连续维度部分检测。由此产生的分级组仍然未知，首席研究员提出了各种方法来揭示这一组的结构。在项目的第二部分，主要研究人员试图通过对西格尔的“椭圆对象”的修改，给出椭圆上同调的几何定义。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，它们都关联着一个希尔伯特空间。在15年多的时间里，由于迈尔-维托里斯原理的失败，西格尔的方法无法转变为上同调理论。新的思想是应用由Connes发展的von Neumann代数的双模的融合，使共形场论在X中局部。当X中对应的圆被分解时，融合被用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。该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，逆关系更常见，其中一个数学概念(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，令人惊讶的数学预测(只有在非常罕见的情况下才能证明)来自理论物理(如量子引力或保形场理论)的考虑。在这个项目的第一部分，首席研究员建议通过冯·诺伊曼最初提出的用于量子力学研究的技术来继续他对4维流形(与相对论最相关)的成功研究。在第二部分中，主要研究者建议提炼共形场理论的概念，以便它可以得到“椭圆上同调”的几何定义。这种上同调在数学上是一个非常成功的工具，所提出的改进有可能导致对所有共形场论的拓扑理解。

DMS-0305280Peter TeichnerIn 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.