Noncommutativity in Low-Dimensional Topology

该项目的首席研究员是德克萨斯州休斯敦威廉·马什·赖斯大学的蒂姆·D·科克伦。该项目的主要目标是寻找非对易代数方法在拓扑学和群论问题中的应用。在过去的十年里，PI和合作者发展了一个所谓的高阶Alexander模的广泛理论，将表格和签名联系起来。它们可以与纽结、链环、3-流形、空间、群甚至曲面同胚相关联。PI和Shelly Harvey在同调如何约束基本群的这些不变量方面得到了许多新的结果。该项目将把这些技术应用于拓扑学和群论中的重要公开问题。具体目标是：找到Heegard Floer纽结同调的一个精化，它更好地反映纽结外部基本群的非交换性；利用高阶签名来构造映射类群的子群的拟同态并构造这样的子群的同调类；进一步研究纽结调和群，包括拓扑性和光滑性；应用这些技巧来研究复2-空间中的代数曲线；继续寻找同调等价与基本群之间的进一步关系，并将这些结果应用于三维流形中的虚拟Betti数问题。这个项目研究三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、物体的卫星识别、人体器官的医学成像和建模以及细胞DNA的功能中都非常重要。尽管所有常见的物体本质上都是三维的，但这样的形状可能会相当复杂。例如，一根缠绕在一起的绳子的形状非常复杂。此外，还有很多事情是未知的：例如，大多数蛋白质的形状。成像设备如何在仅有部分数据的情况下区分坦克和房屋？假设所有的大脑都是不同的，我们如何才能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这类物体复杂的非线性行为的数学思想。小学数学非常线性：2乘以3等于3乘以2。在大学里，人们学到了非对易代数，比如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。这个项目将开发非对易数学的新工具，并将这些工具应用于有关三维物体形状的特定问题。

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.