Graduate Student Colloquium
Louis H. Kauffman
UIC
Introduction to Categorification and Khovanov Homology
Abstract: A classical knot or link is an embedding of a circle or disjoint union of circles in three dimensional space. Classical knot theory is the study of such embeddings up to ambient isotopy, and forms one of the key sources of problems and ideas in geometric and algebraic topology. Mikhail Khovanov in 1999 discovered a remarkable way to associate a chain complex with a diagram of a classical knot or link so that a (graded) Euler characteristic of this homology for the knot K gives the Jones polynomial for K. This Khovanov homology is often referred to as a categorification of the Jones polynomial. Categorification itself refers to creations of larger categories from given categories by taking equalities in a source category and letting them become transformations in a new larger category whose objects are the morphisms in the source category. Part of this talk will be devoted to discussing the meaning of categorification, and part in explaining how the construction of the Khovanov homology works, and how it has been used to give (via Rasmussen's work) a new proof of Milnor's conjecture for the 4-ball genus of torus knots. This talk will be pictorial and self-contained. It helps to have been exposed to homology and the definition of a category before entering the room.
http://en.wikipedia.org/wiki/Homology_(mathematics)
http://en.wikipedia.org/wiki/Category_theory
Friday September 25, 2009 at 3:00 PM in SEO 636