MAT 566, Spring 2024

MAT 566: Differential Topology

Stony Brook Spring 2024

General Course Information, A Proof and a Modern ``Proof"

References

All readings listed in the table below are from Milnor&Stasheff's Characteristic Classes, except

Munkres: Munkres, Elements of Algebraic Topology
G&H: Griffiths&Harris, Principles of Algebraic Geometry
Vakil's slides: Vakil, slides for 2005 MAA Inivited Address
Spanier: Spanier, Algebraic Topology
Atiyah-Bott: Atiyah&Bott, The Moment Map and Equivariant Cohomology, Topology 1984
Mirror Symmetry: K. Hori, et. al., Mirror Symmetry
Milnor's paper: Milnor, On manifolds homeomorphic to the 7-sphere, Annals of Math, 1956
Warner: Warner, Foundations of Differentiable Manifolds and Lie Groups
Spin Geometry: Lawson&Michelsohn, Spin Geometry

Tentative Schedule

Problem Set 5 is due on Monday, 04/09, by 5pm, in Math 3-111

Date	Topic	Read	Problem Set
01/23, Tu	Review/overview of vector bundles	Sections 1-3; VB Notes	ps1
01/25, Th	Review of (co)homology	Munkres: Sections 31,33 Appendix A: pp257-260, 263bot-270
01/30, Tu	Review of (co)homology	Munkres: Sections 31,33 Appendix A: pp257-260, 263bot-270
02/01, Th	Poincare Duality	Appendix A: pp270-279
02/06, Tu	Stiefel-Whitney classes	Section 4: pp37-50	ps2
02/08, Th	Stiefel-Whitney classes and cobordism Grassmannians	Section 4: pp50-53 Section 5: pp55-70; G&H: pp193-194,207
02/15, Th	(Co)homology of CW-complexes	Sect. 6: pp73-74; App. A: pp260bot-263
02/20, Tu	(Co)homology of Grassmannians	Sections 6,7; G&H: pp194-197
02/22, Th	Schubert Calculus	Vakil's slides; notes; G&H: pp197-206	ps3
02/27, Tu	Thom Isomorphism Theorem	Spanier pp255-263; Munkres: pp279-280 Sections 8,10; App. A: pp265bot-268top	ps3
03/05, Th	Applications of TI	Section 9	ps4
03/07, Tu	Euler class	Section 9
03/19, Tu	Topology of smooth manifolds	Section 11
03/21, Th	Topology of smooth manifolds	Section 11
03/26, Tu	Chern classes	Section 12: pp143bot-146 Sections 13,14	ps5
03/28, Th
04/02, Tu
04/04, Th	Atiyah-Bott Localization Theorem	Atiyah-Bott's paper: Sections 2,3 Notes on Mirror Symmetry: Chapter 1
04/09, Tu	Pontrjagin classes	Section 15	ps6
04/11, Th	Chern and Pontrjagin numbers	Section 16: pp183-193
04/16, Tu	Oriented cobordism ring	Section 16: pp190-195; Section 17
04/18, Th	Some homotopy theory	Spainer: Sections 7.2-7.5	ps7
04/23, Tu	Thom-Pontrjagin construction	Section 18
04/25, Th	Multiplicative sequences	Section 19
04/30, Tu	Hirzebruch Signature Theorem	Section 19
05/02, Th	Exotic smooth structures on S⁷	Milnor's paper	none

This page is maintained by Aleksey Zinger.
Last modified: April 18, 2024.