Topics include the Whitney trick and its failure in 4 dimensions; its analogues in 3-dimensions: the Dehn lemma and the loop theorem; Milnor's invariants for classical links; various special 2-complexes important if 4-manifold topology: Casson towers, Whitney towers, capped gropes; Heegaard splittings; Khovanov homology and other aspects of categorication.
The main goal of the class is to give an introduction to geometric and quantum topology in low (2, 3, 4) dimensions. Specific topics that will be discussed include the (colored) Jones polynomial of knots, the Jones-Wenzl projectors, quantum invariants of 3- manifolds, Kirby calculus, applications to the chromatic polynomial of planar graphs.
This class gives an overview of both classical and modern results in Knot theory: 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.
Topics include: manifolds, the Gauss map, parallel transport, curvature of curves and surfaces, the first and second fundamental forms, surface area and volume, geodesics, the exponential map, the Gauss-Bonnet theorem.