Noncommutativity in Low-Dimensional Topology

William Marsh Rice University, Houston TX

Investigators

Abstract

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.