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.
In its broadest terms this project investigated the mathematical structure of the ``shape'' of 3 and 4-dimensional objects. Understanding complex shapes has many applications. The geometric structure of proteins and other complex molecules is crucial to drug development. The geometric configuration of cellular DNA is important in determining the precise mechanisms of cellular processes. The geometric shape of organs is vital to the field of medical imaging. The prediction of shape from incomplete data (satellite imaging, sensor networks) has important military and commercial applications. Additionally, this project increased the participation of U.S. citizens, permanent residents and women in mathematical research and education. The project focussed on knot theory, a study which is known to parametrize all 3-dimensional objects; and especially on the equivalence relation of knot concordance, which is known to be closely related to 4-dimensional objects. Specifically this project established a new framework, called the bipolar filtration, that allows researchers to quantify the difference between topological disks and disks that are smoothly embedded. Counterexamples were found to a 30-year old conjecture of Louis Kauffman concerning which knots can be unknotted in 4 dimensions. This discovery will drastically alter the future path of this reseacrh area. This project demonstrated that the set of knot concordance classes can be seen as a fractal space, that is it was shown to be a metric space with many natural self-similarities, called satellite operations, and some of these operations were shown to be quasi-isometric embeddings and some to be quasi-contractions. It was also shown that neither the concordance class of a knot , nor its tau-invariant nor s-invariant is determined by its zero-framed surgery. The project supported 8 mathematics PhD students, including 6 US citizens, 2 women, 1 and one Hispanic. All of the 13 research publications and preprints that appeared during the project period were made immediately available to other researchers on a widely-used preprint server. The PI collaborated with 9 researchers, including 4 women and 6 persons within 5 years of their PhD. During the project period the PI mentored 6 postdoctoral instructors.
