A fundamental goal in low-dimensional topology is the classification of manifolds. The fundamental group of the manifold is an important algebraic invariant, but is rarely abelian. On the other hand, since the homology groups of the manifold are abelian, they can be classified easily. Unfortunately, a lot of the rich structure of the fundamental group is lost when considering the homology groups. Homology with local coefficients associates modules to the manifold. Since modules are abelian groups, distinguishing them is still tenable. Furthermore, since these are often modules over noncommutative rings, some of the rich structure of the fundamental group is retained. These homology modules, as well as linking forms defined on them, have been used to define algebraic invariants of knot complements, 3-manifolds, and algebraic curve complements. These have led to new results about knot concordance, obstructions to the existence of symplectic structures on the product of a 3-manifold with the circle, and restrictions on which groups can be realized as the fundamental group of an algebraic curve complement. The goal of this project is to use these noncommutative techniques to find answers to questions related to three particular areas of low-dimensional topology: knot Floer homology, the topology of algebraic curve complements, and the structure of the knot concordance group.
The goal of this project is to use non-commutative algebra to better understand knotted curves and surfaces. The study of knotted curves and surfaces has important applications in biology and physics. For example, DNA strands are naturally knotted, but must unknot in order to replicate. Also understanding knotted surfaces is fundamental to understanding the shape of the universe. Many mathematicians have used algebra to better understand how curves and surfaces can knot, however the type of algebra considered is usually commutative. By employing non-commutative algebras, we can get a more refined understanding for how curves and surfaces can knot.
