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
12/5/23