Principal Investigator: Jozef H. Przytycki
This award provides partial support for participants of the 21st conference in the very successful ``Knots in Washington'' series. This meeting is devoted to Skein Modules, Khovanov Homology and its relation to Hochschild Homology. ``Khovanov homology'' is a new development in quantum topology that emerged in the last several years. It is a generalization of the Jones and Homflypt polynomials of links to homologies of certain chain complexes. These homologies turn out to be significantly stronger than the original invariants. Khovanov's ideas were also applied to the polynomial invariants of graphs, as well as Kauffman bracket skein module of some 3-manifolds. One of the most recent developments is Przytycki's observation that Khovanov homology (or its comultiplication-free variant developed by Helme-Guizon and Rong) can be interpreted as Hochschild homology of underlying algebras. This shows how seemingly distant branches of mathematics can be put together.
Low dimensional topology studies shapes of three and four dimensional spaces. These dimensions are of particular interest to us because they are the dimensions of our space and our space-time. Knot theory is a subfield of low dimensional topology, which studies the knottedness in our space. One of the main goals of knot theory is to distinguish knotted objects. This is often done by means of the so called "knot invariants," functions that replace geometric objects, such as knots, with those that are easier to compare. Over the past two decades, there have been a flourish of new invariants in low dimensional topology, boosted by ideas from gauge theory, quantum algebras, and mathematical physics. In particular, a new invariant for knots, developed by Khovanov using ideas from homological algebra, has sparked a great deal of interest recently. The purpose of the Knots in Washington Conferences is to bring together the researchers of the field, including established mathematicians as well as graduate students and recent PhDs, to discuss the state of the art of the subject.
