Gauge theory and smooth invariants
Winter 2015 & Winter 2013
Dr. Ben Mares
$2\,g(\Sigma)-2 \, \geq \, \left|\left\langle c_1(\mathfrak s),\,\left[\Sigma\right]\right\rangle\right|+\left[\Sigma\right]\cdot\left[\Sigma\right]$
(the adjunction inequality)
Location and time
Room 136, from Thursday 29 January, Tuesdays 9:45-11:00 and Thursdays 11:00-12:30.Announcements
- Second problem set posted, due Thursday 12th.
- I always appreciate constructive feedback about my teaching via this anonymous survey.
Lecture notes
Draft in progressProblem sets
Old stuff
Course Outline
Problem sets
Notes
- Lecture 1: Overview
- Lecture 2: Intro to cohomology, smooth structures
- Lecture 3: Classification in dimensions 0, 1, 2, and intersection forms
- Lecture 4: Functorial classification and chirality of $\mathbb{CP}^2$
- Lecture 5: More levels of classification and examples; singular cohomology
- Lecture 6: General coefficients and homology
- Lecture 7: Poincaré duality and representing homology
- Lecture 8: Intersection theory, and classification results
- Lecture 9: Classification of smooth simply connected 4-manifolds up to homeomorphism, and Čech cohomology
- Lecture 10: Bundle theory and generalized Čech cohomology
- Lecture 11: Non-abelian Čech cohomology, and extension of structure group
- Lecture 12: Reduction of structure group, classification of compact simple Lie groups
- Lecture 13: Classification of principal bundles, Hodge star
- Lecture 14: Intro to Hodge decomposition
- Lecture 15: Proof of Hodge decomposition, up to elliptic theory
- Lecture 16: Introduction to Sobolev spaces and distributions
- Lecture 17: Elliptic operators and proofs of embedding theorems on $T^n$
- Notes on Sobolev multiplication
Links
- MathOverflow discussion of Classification of 1-dimensional manifolds (not second-countable), and articles about exotic structures on the long line and the long plane, thanks to Giacomo.