CAREER: Algebraic Methods in Low-Dimensional Topology

职业：低维拓扑中的代数方法

• 批准号: 0748458
• 负责人: Shelly Harvey
• 金额: $ 44.33万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-05-15 至 2014-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0748458&HistoricalAwards=false
• 关键词: CAREER; Algebraic; Methods; Low; Dimensional

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.

该项目旨在利用几何、拓扑和代数技术的组合来解决 3 维流形、结理论、映射类群和接触拓扑领域的具体问题。泛函分析和冯·诺依曼代数的工具也正在被使用。 PI拟承担以下项目。 (1)在结索引组的分类方面取得重大进展。 特别是，对其 (n) 可解过滤的连续商进行分类，并在群中找到新的结构。 (2)建立高阶Heegaard Floer同调理论，对PI定义的高阶亚历山大多项式进行分类。 用它来表明拓扑切片结的某些经典族不是平滑切片的。 (3)定义与广义约翰逊子群相关的映射类群的新的有趣的规范子群，并证明它们的同调群是无限生成的。 (3)确定曲面映射类群的某些子群与其映射环面(3流形)的拓扑之间的精确关系。 (4) 了解 S^3 中的横向结与作为循环和简单分支覆盖层出现的任意 3-流形的接触结构之间的精确关系。 利用这种关系可以更好地理解接触结构的几何不变量，例如支撑亏格和结合数。理解 3 维空间中物体的几何结构具有至关重要的科学意义。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，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3 维物体的精确数学描述。当人们想到精确的数学描述时，人们通常会用数字来思考，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，二十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，需要向量和矩阵，并且矩阵乘法不可交换，即 AB 通常不等于 BA。 因此，每个物理交互都基于非交换代数。 该项目正在研究这种非交换代数如何产生 3 维空间和 3 维空间中的对象的几何结构的数学描述。 特别重要的是 3 维空间中的闭合弦在 3 维和 4 维中打结的方式。 PI 将使用非交换数学对象来更好地理解字符串和 3 维空间的一般结点。