Math 4750: Introduction to Knot Theory

Knot theory is a classical subject concerning knotting and linking in 3-space. Many new and interesting developments took place in this subject over the last 30 years. This class will give an overview of both classical and recent results: the Kauffman bracket and the Jones polynomial; the Alexander polynomial; Khovanov homology; Seifert surfaces; examples and classes of knots: torus knots, alternating knots, hyperbolic knots; intrinsic knotting of graphs; colorings and relation to combinatorics of planar graphs; knotting and linking in higher dimensions.

Math 7830: Fiber Bundles

Course material: Vector bundles, principal bundles, characteristic classes (Stiefel-Whiteney, Chern, Pontryagin). Orientability, Euler class. Classifying spaces (for vector bundles and principal bundles). Spectral sequences: Leray-Serre and applications. Elements of K-theory.