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.

该项目旨在使用几何，拓扑和代数技术的组合在3维流形，结理论，映射课程组和接触拓扑领域的特定问题中解决特定问题。功能分析和von Neumann代数的工具也正在使用。 PI建议进行以下项目。 （1）在结的分类方面取得了重大进步。 特别是，对其（n）可分化过滤的连续商进行分类，并在组中找到新的结构。 （2）建立高阶Heegaard浮子同源理论，该理论分类了PI定义的高阶亚历山大多项式。 用它表明某些经典的拓扑切成薄片家族不是平稳的切片。 （3）定义与广义约翰逊亚组相关的映射类组的新有趣的规范子组，并表明其同源组是无限生成的。 （3）确定表面映射类组的某些亚组与映射托里的拓扑（3个manifolds）之间的精确关系。 （4）理解S^3中的横向结与任意3个manifolds的接触结构之间的精确关系，这些关系是循环和简单的分支覆盖物。 利用这种关系来更好地了解接触结构的几何不变，例如支持属和结合数。理解在三维空间中对象的几何结构是至关重要的科学重要性。从基于细胞DNA的打结到基于蛋白质的几何形状的抗病毒药物的癌症治疗，到心脏形状的非侵入性可视化，再到考虑时空本身的“形状”，我​​们寻求3维对象的精确数学描述。当人们想到一个精确的数学描述时，人们通常会在数字上思考，但是普通数字不足以捕获我们世界的复杂性。 普通数字的乘法是``可交换''。但是，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，并且矩阵的乘法不是交换性的，即AB通常不等于Ba。 因此，每种物理相互作用都是基于非共同代数的。 该项目正在研究该非共同代数如何产生3维空间的几何结构和3维空间中对象的几何结构的数学描述。 特别重要的是，在3维空间中的封闭串在3-维度和4维中打结。 PI将使用非交互性数学对象来更好地理解字符串和三维空间的打结。