Teaching

MATH 8750: Topology of Manifolds

Semester: 

Fall

Offered: 

2019
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.

MATH 7830: Fiber Bundles

Semester: 

Spring

Offered: 

2019
Course material: Vector bundles, principal bundles, associated 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.

MATH 8750: Topology of Manifolds

Semester: 

Fall

Offered: 

2018
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.

MATH 4750: Introduction to Knot Theory

Semester: 

Spring

Offered: 

2018
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.

MATH 4720: Introduction to Differential Geometry

Semester: 

Spring

Offered: 

2017
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.

MATH 3340: Complex Variables

Semester: 

Fall

Offered: 

2016
Topics covered in this class include analytic functions, complex integration, Cauchy formulas, power series, residue theory, conformal mappings, and Laplace transforms.

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