Noncommutativity in Low-Dimensional Topology

低维拓扑中的非交换性

• 批准号: 0706929
• 负责人: Tim Cochran
• 金额: $ 28.29万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706929&HistoricalAwards=false
• 关键词: Noncommutativity; Low; Dimensional; Topology

The Principal Investigator of the project is Tim D. Cochran of William Marsh Rice University in Houston Texas. The broad goal of the project is to find applications of methods of noncommutative algebra to problems in topology and group theory. Over the last ten years the PI and collaborators have developed a vast theory of so-called higher-order Alexander modules, linking forms and signatures. These can be associated to knots, links, 3-manifolds, spaces, groups or even surface homeomorphisms. The PI and Shelly Harvey have many new results on how homology constrains these invariants of the fundamental group. The project will apply these techniques to important open problems in topology and group theory. Specific goals are: to find a refinement of Heegard Floer Knot Homology that better reflects the noncommutativity of the fundamental group of the knot exterior; to use higher-order signatures to construct quasi-homomorphisms of subgroups of mapping class groups and to construct homology classes for such subgroups; to further investigate the knot concordance group, both topological and smooth; to apply these techniques to study algebraic curves in complex 2-space; to continue to find further relationships between homology equivalence and fundamental group and apply these results to the virtual betti number problem in 3-manifolds. This project studies mathematical aspects of the shape, or topology, of 3-dimensional objects. Shape is very important in the study of networks, search algorithms, in the design of drugs, satellite recognition of objects, the medical imaging and modeling of human organs and in the function of cellular DNA. Even though all common objects are 3-dimensional in nature, such shapes can be quite complicated. For example, the shape of a tangled piece of string is quite complex. Moreover much is unknown: the shapes of most proteins, for example. How can an imaging device distinguish a tank from a house given only partial data? How can one usefully quantify the shape of a brain given that all brains are different? The scientific study of shape requires mathematical ideas that can accurately quantify the complex non-linear behavior of such objects. Grade-school mathematics is very linear: 2 times 3 equals 3 times two. In college one learns that noncommutative algebra, such as matrices where AB is not necessarily BA, is necessary to model simple real-life situations. This project will develop new tools in noncommutative mathematics and apply these to specific problems concerning the shape of 3-dimensional objects.

该项目的首席研究员是德克萨斯州休斯敦威廉马什莱斯大学的蒂姆·D·科克伦(Tim D. Cochran)。该项目的总体目标是寻找非交换代数方法在拓扑和群论问题中的应用。在过去的十年里，PI 和合作者开发了一个庞大的所谓高阶亚历山大模块理论，连接表格和签名。这些可以与结、链接、三流形、空间、群甚至表面同胚相关。 PI 和 Shelly Harvey 关于同源如何约束基本群的这些不变量有许多新的结果。 该项目将应用这些技术来解决拓扑和群论中重要的开放问题。具体目标是：找到 Heegard Floer 结同调的改进，更好地反映结外部基本群的非交换性；使用高阶签名来构造映射类群的子群的拟同态，并为这样的子群构造同源类；进一步研究结语索引群，无论是拓扑还是平滑；应用这些技术来研究复二维空间中的代数曲线； 继续寻找同源等价与基本群之间的进一步关系，并将这些结果应用于 3 流形中的虚拟贝蒂数问题。该项目研究 3 维物体的形状或拓扑的数学方面。形状在网络研究、搜索算法、药物设计、卫星物体识别、人体器官的医学成像和建模以及细胞 DNA 功能中非常重要。尽管所有常见物体本质上都是 3 维的，但此类形状可能相当复杂。例如，一根缠结的绳子的形状就相当复杂。此外，还有很多未知的事情：例如，大多数蛋白质的形状。在仅给出部分数据的情况下，成像设备如何区分坦克和房屋？鉴于所有大脑都是不同的，如何有效地量化大脑的形状？对形状的科学研究需要能够准确量化此类物体复杂的非线性行为的数学思想。小学数学是非常线性的：2乘以3等于3乘以二。在大学里，人们了解到非交换代数（例如 AB 不一定是 BA 的矩阵）对于模拟简单的现实生活情况是必要的。该项目将开发非交换数学的新工具，并将其应用于有关 3 维物体形状的特定问题。