CAREER: Algebraic Methods in Low-Dimensional Topology

本项目旨在利用几何、拓扑和代数技术的结合，解决三维流形、结理论、映射类群和接触拓扑等领域的具体问题。从泛函分析和冯·诺伊曼代数的工具也被使用。PI建议开展以下项目：(1)在结谐和群分类方面取得重大进展。特别地，对其(n)可解过滤的连续商进行分类，并在群中发现新的结构。(2)建立了对PI定义的高阶Alexander多项式进行分类的高阶Heegaard flower同调理论。用它来证明某些经典的拓扑切片结族不是平滑切片的。(3)定义与广义Johnson子群相关的映射类群的新的有趣正则子群，并证明它们的同调群是无限生成的。(3)确定曲面的映射类群的某些子群与其映射环面的拓扑（即3流形）之间的精确关系。(4)理解S^3中的横向结与任意3流形的接触结构之间的精确关系，这些流形以循环和简单分支盖的形式出现。使用此关系可以更好地理解接触结构的几何不变量，如支持属和绑定数。了解三维空间中物体的几何结构具有重要的科学意义。从基于细胞DNA打结的癌症治疗，到基于蛋白质几何形状的抗病毒药物，再到心脏形状的非侵入性可视化，再到思考时空本身的“形状”，我们寻求对三维物体的精确数学描述。当人们考虑一个精确的数学描述时，人们通常会想到数字，但普通的数字不足以捕捉我们世界的复杂性。普通数字的乘法是可交换的。“然而，20世纪的物理学告诉我们，物质和能量不能仅仅用数字来描述。相反，向量和矩阵是必需的，矩阵的乘法是不可交换的，也就是说AB通常不等于BA。因此，每个物理相互作用都是基于非交换代数的。这个项目是研究这种非交换代数如何产生三维空间和三维空间中物体的几何结构的数学描述。特别重要的是三维空间中的闭弦在三维和四维中打结的方式。PI将使用非交换数学对象来更好地理解一般的弦和三维空间的打结。

This project seeks to resolve specific questions in the area of 3- dimensional manifolds, knot theory, mapping class groups, and contact topology using a combination of geometric, topological and algebraic techniques. Tools from functional analysis and von Neumann algebras are also being used. The PI proposes to undertake the following projects. (1) Make significant advances toward the classification of the knot concordance group. In particular, classify the successive quotients of its (n)-solvable filtration and find new structure in the group. (2) Establish a higher-order Heegaard Floer homology theory that categorifies the higher-order Alexander polynomials defined by the PI. Use this to show that certain classical families of topologically slice knots are not smoothly slice. (3) Define new interesting canonical subgroups of the mapping class group related to the generalized Johnson subgroups and show their homology groups are infinitely generated. (3) Determine the precise relationship between certain subgroups of the mapping class group of a surface and the topology of their mapping tori (which are 3-manifolds). (4) Understand a precise relationship between transverse knots in S^3 and contact structures of arbitrary 3-manifolds that arise as cyclic and simple branched covers. Use this relationship to better understand the geometric invariants of a contact structure such as the support genus and binding number.Understanding the geometric structure of objects in 3-dimensional space is of crucial scientific importance. From cancer treatments based on the knotting of cellular DNA, to antiviral drugs based on the geometrical shapes of proteins, to non-invasive visualization of the shape of the heart, to contemplating the ``shape'' of space-time itself, we seek precise mathematical descriptions of 3-dimensional objects. When one thinks of a precise mathematical description, one often thinks in terms of numbers, but ordinary numbers are insufficient to capture the complexities of our world. Multiplication of ordinary numbers is ``commutative.'' However, the physics of the twentieth century has taught us that matter and energy cannot be described merely by numbers. Rather, vectors and matrices are required, and multiplication of matrices is not commutative, that is AB does not usually equal BA. Every physical interaction is thus based on noncommutative algebra. This project is investigating how this noncommutative algebra yields a mathematical description of the geometric structure of 3-dimensional space and of objects in 3- dimensional space. Of particular importance is the manner in which closed strings in 3-dimensional space are knotted in 3- and in 4- dimensions. The PI will use non-commutative mathematical objects to better understand the knotting of strings and 3-dimensional spaces in general.