Noncommutative algebraic invariants in topology

该项目的首席研究员是Tim D。他是德克萨斯州休斯敦市威廉·马什·赖斯大学的教授。该项目的广泛目标是找到非交换代数方法在拓扑学、群论和数论问题中的应用。在过去的12年里，PI和合作者开发了一个所谓的高阶亚历山大模块的庞大理论，连接形式和签名。这些可以与结，链接，3-流形，空间，群甚至表面同胚相关联。 该项目将应用这些技术在拓扑和群论的重要开放问题。具体目标是：在光滑纽结和链环协调群中寻找结构，特别是在拓扑切片纽结的光滑协调类的子群中寻找结构，证明纽结协调群的COT滤子项的准素分解定理，在双曲三维流形的同调配边类中寻找相似的结构和复杂性，在拓扑切片纽结的光滑协调类中寻找结构，在拓扑找到Heegaard Floer Knot Homology的一个改进，它更好地反映了Knot外部基本群的非交换性;使用类似的技术找到映射类群的子群的新的同调元素;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Betti数问题。PI还将扩展，亲p组的类别，Stallings，德怀尔，和Cochran-Harvey相关的组同源性的基础结果导出和较低的中心series.This项目研究的数学方面的形状，或拓扑结构，三维物体。形状对于抗病毒药物的设计，网络研究，搜索算法，物体的卫星识别，人体器官的医学成像和建模以及细胞DNA的功能至关重要。尽管所有常见的物体本质上都是三维的，但这些形状可能相当复杂。例如，蛋白质等大分子的形状非常复杂，而且大多数都是未知的，尽管这些知识对于创造抗病毒药物至关重要。成像设备如何在仅给出部分数据的情况下区分坦克和房屋？既然所有的大脑都是不同的，那么如何有效地量化人类大脑的形状呢？形状的科学研究需要数学思想，可以准确地量化这些对象的复杂非线性行为。非交换代数，例如AB不一定等于BA的矩阵，对于模拟简单的现实生活情况是必要的。这个项目将在非交换数学中开发新的工具，并将其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, group theory and number theory. Over the last 12 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 project will apply these techniques to important open problems in topology and group theory. Specific goals are: to find structure in the smooth knot and link concordance groups, in particular to find structure in the subgroup of smooth concordance classes of topologically slice knots; to prove primary decomposition theorems for terms of the COT filtration of the knot concordance group; to find similar structures and complexity in the homology cobordism classes of hyperbolic 3-manifolds; to find a refinement of Heegaard Floer Knot Homology that better reflects the noncommutativity of the fundamental group of the knot exterior; to use similar techniques to find new elements of homology for subgroups of mapping class groups; 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. The PI also will extend, to the category of pro-p groups, the foundational results of Stallings, Dwyer, and Cochran-Harvey relating group homology to derived and lower central series.This project studies mathematical aspects of the shape, or topology, of 3-dimensional objects. Shape is crucial to the design of antiviral drugs, the study of networks, search algorithms, 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 large molecule, such as a protein is quite complex, and mostly unknown, despite the fact that such knowledge is vital to the creation of antiviral drugs. How can an imaging device distinguish a tank from a house given only partial data? How can one usefully quantify the shape of a human 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. Noncommutative algebra, such as in matrices where AB is not necessarily equal to 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.