Welcome to the cosmos of Low Dimensional Topology. This is the area of mathematics that deals with the study of smooth manifolds of dimensions < 5. We will explore the land of exotic shapes and geometries that arise from these low dimensional curiosities. Somehow, the dimensional constraints offered by these objects (such as knots, surfaces, 3 and 4-manifolds) manifest some of the most interesting geometric phenomena.
While certainly beautiful, the downside to this subject is that it requires a lot of machinery to even define (and work with) these objects. The purpose of this course is to provide an overview of some essential ideas in point-set topology, differential topology, and algebraic topology and expose students to the aromas and flavors of low dimensional topology before they might usually encounter them. That being said, this class is NOT meant to be a replacement for more structured classes in geometry and topology (e.g. Math 140, 141, 142, 202a, 214, 215a/b, etc.), but we hope to be a helpful supplement to motivate and introduce a lot of the concepts taught more in depth in said classes and, moreover, touch on some applications that are typically withheld until further study.
Week 1 (8/23 - 25): The trefoil is a fibered knot (No official reading)
Wed: logistics, Intro, Example of fibering trefoil
Fri: Finish example of fibering trefoil
Week 2 (8/28-9/1): Begin point-set
Mon: Motivation from metric spaces (104 review in context of topology)
Wed: Axioms of topological space, Continuity
Fri: Basis of a Topology. Product and Quotient Topologies (key example: RP^2)
Week 3 (9/4-9/8): Cont. Point-set
Mon: Labor Day, NO CLASS
Wed: Finish quotient topology + Compactness
Fri: Finish Compactness
Week 4 (9/11-9/15): Finish Point-set
Mon: Connectedness
Wed: Topological Manifolds intro
Fri: Mapping Class Groups
Week 5 (9/18-9/22): Start Manifold Theory
Mon: Smooth Manifolds and Manifolds with Boundary
Wed: Smooth Manifolds
Fri: Smooth Manifolds
Week 6 (9/25-9/29): Algebraic Topology
Mon: Homotopy
Wed: Fundamental Group
Fri: Fundamental Group of S^1 and Wirtinger Presentation
Week 7 (10/2-10/6): Morse Theory
Mon: Motivate Morse theory + Morse Lemma
Wed: Finish Morse lemma and start flows
Fri: Flows and critical points
Week 7 (10/9-10/13): Introduce Knot theory
Mon: Basic definitions + elementary knot invariants
Wed: Alexander Polynomial
Fri:
Week 8 (10/16-10/20): Topics + Knot theory
Mon: Hopf Fibration
Wed: Generalized Homology Theories
Fri: Jones Polynomial
Week 9 (10/23-10/27): Topics + Knot theory
Mon: Jones Polynomial Cont.
Wed: (Dehn) Surgery
Fri: Fundamental Theorem of Lickorish and Wallace
Week 10 (10/30-11/3): Topics
Mon: Handlebody Theory
Wed: Spun Knots
Fri: Fundamental Theorem of Lickorish and Wallace
Week 11 (11/6-11/8): Guest Lectures
Mon: Sard's Theorem
Wed: Sard's Theorem cont.
Fri: Holiday
Week 12: (11/13-11/17): Special Topics
Mon: Riemann Surfaces and the Uniformization theorem
Wed: Introduction to TQFTs
Fri: TQFTs cont. Dijkgraaf Witten theory.
Week 13: Holiday
Week 14: (11/27-12/1): Special topics
Mon: Finish Fundamental Theorem of Lickorish and Wallace
Wed: Planar Folding and Fold Invariants of Knots
Fri: Introduction to Topological Data Analysis