CAREER: Algebraic Methods in Low-Dimensional Topology

这个项目寻求使用几何、拓扑和代数技术的组合来解决三维流形、纽结理论、映射类群和接触拓扑学领域的具体问题。泛函分析和von Neumann代数的工具也被使用。国际和平研究所建议开展以下项目。(1)在纽结和谐组的分类方面取得重大进展。特别地，对其(N)-可解滤子的连续商进行了分类，并在群中找到了新的结构。(2)建立了高阶Heegaard Floer同调理论，对PI定义的高阶Alexander多项式进行了分类。使用它来表明某些经典拓扑片纽结不是光滑切片的。(3)定义了与广义Johnson子群相关的映射类群的新的有趣的正则子群，并证明了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(3-流形)的拓扑之间的精确关系。(4)理解S[3]中的横向纽结与任意三维流形的接触结构之间的精确关系，这些结构作为循环和单分支覆盖出现。利用这种关系可以更好地理解接触结构的几何不变量，如支承亏格和结合数。理解三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，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”，我们寻求对三维对象的精确数学描述。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反映我们世界的复杂性。普通数的乘法是“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，向量和矩阵是必需的，而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建立在非对易代数的基础上。这个项目正在研究这个非对易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3维空间中闭合弦在3维和4维空间中打结的方式。PI将使用非交换的数学对象来更好地理解弦和一般3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.