MTH 782: Topics in Topology
University of Miami, Fall 2023
Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: 8:15-9:15 Tuesday, 11-12 Thursday, or by appointment
Time and Location: TuTh 12:30PM - 1:45PM, Ungar 406
Course syllabus can be found here.
There is no single textbook for the course. References for some material covered:
[DK] The Geometry of Four-manifolds by Donaldson and Kronheimer
[Mor] The Seiberg-Witten Equations and Applications to the Topology of Smooth 4-Manifolds by Morgan
[Moo] Lecture Notes on Seiberg-Witten Invariants by Moore
[KM] The Genus of Embedded Surfaces in the Projective Plane by Kronheimer and Mrowka
[GS] 4-manifolds and Kirby Calculus by Gompf and Stipsicz
[AHS] Self-Duality in Four-Dimensional Riemannian Geometry by Atiyah, Hitchin, Singer
[D] Self-dual connections and the topology of smooth 4-manifolds by Donaldson
[Dbm] Nahm's equations and the classification of monopoles by Donaldson
[AH] The Geometry and Dynamics of Magnetic Monopoles by Atiyah and Hitchin
[FS] The blowup formula for Donaldson invariants by Fintushel and Stern
[FS2] Knots, links, and 4-manifolds by Fintushel and Stern
[KM1] Embedded surfaces and the structure of Donaldson's polynomial invariants by Kronheimer and Mrowka
[KM2] Monopoles and Three-Manifolds by Kronheimer and Mrowka
[D2] Floer Homology Groups in Yang-Mills Theory by Donaldson
[S] Notes for the course written by myself. (These will be emailed.)
Course Schedule
Date |
Lecture Content |
Reading |
8/22/23 |
Introduction. Basic invariants of 4-manifolds. Lattices. |
[DK] 1.1, 1.2, [S] |
8/24/23 |
Freedman's Classification. Rohlin's Theorem. |
[S] |
8/29/31 |
Statement of Donaldson's Theorem. Big open questions. |
[S] |
8/31/23 |
Basics of Characteristic classes. Hirzebruch Signature Theorem. |
[GS] Ch.1 |
9/5/23 |
Differential forms. Hodge theorem. |
[DK] Ch.1, [S] |
9/7/23 |
(anti-)self-dual forms. ASD complex. the period map. |
[S] |
9/12/23 |
Bundles and connections. |
[DK] Ch.2, [S] |
9/14/23 |
New connections from old. Gauge group. Parallel transport. |
[DK] Ch.2, [S] |
9/19/23 |
Curvature. |
[DK] Ch.2, [S] |
9/21/23 |
Flat connections and holonomy. Chern Weil theory. |
[DK] Ch.2, [S] |
9/26/23 |
Yang-Mills functional. Instantons. |
[DK] Ch.2, [S] |
9/28/23 |
The basic instanton. Instantons on S4. |
[DK] 3.3, 3.4, [S] |
10/3/23 |
Moduli theory 1. |
[DK] Ch.4 (especially 4.2, 4.3), [S] |
10/5/23 |
Moduli theory 2. |
[DK] Ch.4, [S] |
10/10/23 |
Atiyah-Hitchin-Singer Theorem. |
[AHS], [S] |
10/12/23 |
Proof of Donaldson's Diagonalization Theorem. |
[D], [DK] Ch.8 (8.1) |
10/17/23 |
Fall Recess |
|
10/19/23 |
Some more details on reducibles. Donaldson invariants. |
|
10/24/23 |
First applications of Donaldson invariants. |
[DK] Ch.9 |
10/26/23 |
Remarks U(2) bundles. Dimensional reductions. Bogomolny Monopoles. |
[AH] |
10/31/23 |
More BMs. Vanishing results for Donaldson invariants. |
[Dbm] |
11/2/23 |
The blowup formula for Donaldson invariants. |
[FS] |
11/7/23 |
Kronheimer and Mrowka's structure theorem. |
[KM1] |
11/9/23 |
Intro to Seiberg-Witten theory. |
[KM2] Ch 1, see also [Moo], [Mor] |
11/14/23 |
SW invariants and Witten's Conjecture. |
[KM2] Ch 1 |
11/16/23 |
Adjunction inequalities from SW theory. Fintushel Stern knot surgery. |
[KM2] 40.2, [FS2] |
11/21/23 |
Thanksgiving Recess |
|
11/23/23 |
Thanksgiving Recess |
|
11/28/23 |
Intro to Morse homology. The Chern-Simons functional. |
[KM2] Ch 1, [D2] Ch 1 |
11/30/23 |
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12/5/23 |
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