MAT 566, Fall 2006

MAT 566: Differential Topology

Stony Brook Fall 2006

Announcements

Problem Set 7 is due on Thursday, December 14, by 5pm, in Math 3-117.

Here are some Notes on the Splitting Principle .

Here is general information about the course (revised 9/7).

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
Expository Notes: Zinger, Counting Plane Rational Curves
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

Course Instructor

Name: Aleksey Zinger E-mail: azinger@math Phone: 432-8618
Office: Math Tower 3-117 Office Hours: Wed 9-12

Schedule

Problem Set 7 is due on Thursday, December 14, by 5pm, in Math 3-117

Date	Topic	Read	Problem Set
9/7	Review of Vector Bundles	Sections 1-3	pdf, ps
9/12	Review/Overview of (Co)homology	Munkres: Chap 7; Sects 31,33 App. A: pp257-260, 263bot-266, 268
9/14	Review/Overview of (Co)homology
9/19	(Co)homology of Manifolds	Appendix A: pp270-279
9/21	Applications of Stiefel-Whitney Classes	Section 4: pp37-50
9/26	Stiefel-Whitney Classes and Cobordism Grassmannians	Section 4: pp50-53 Section 5: pp55-62; G&H: pp193-194,207	pdf, ps
9/28	Grassmannians and Vector Bundles	Section 5: pp62-70
10/3	(Co)homology of CW-Compelexes	Sect. 6: pp73-74; App. A: pp260bot-263
10/5	(Co)homology of Grassmannians	Sections 6,7; G&H: pp194-197
10/10	Schubert Calculus (by Paul and Ritwik)	G&H: pp197-206; Vakil's slides
10/12	Thom Isomorphism Theorem	Spanier pp255-263; Munkres: pp279-280 Sections 8,10; App. A: pp265bot-268top	pdf, ps
10/17	Thom Isomorphism Theorem		pdf, ps
10/19	Euler Class	Section 9	pdf, ps
10/24	Topology of Smooth Manifolds	Section 11
10/26	Topology of Smooth Manifolds	Section 11
10/31	Chern Classes	Section 12: pp143bot-146 Sections 13,14	pdf, ps
11/2	Chern Classes	Section 12: pp143bot-146 Sections 13,14
11/7	Enumerative Geometry of Plane Curves (by Caner and Chris)	expository notes
11/9	Atiyah-Bott Localization Theorem (by Robert F. and Thomas)	Atiyah-Bott's paper: Sections 2,3 Mirror Symmetry: Sections 4.1-4.3
11/14	Pontrjagin Classes	Section 15	pdf, ps
11/16	Chern and Pontrjagin Numbers	Section 16: pp183-193
11/21	Applications to Oriented Cobordism Ring	Section 16: pp190-195; Section 17
11/28	Some Homotopy Theory	Spainer: Sections 7.2-7.5 Section 18	pdf, ps
11/30	Thom-Pontrjagin Construction	Spainer: Sections 7.2-7.5 Section 18
12/5	Multiplicative Sequences	Section 19: pp219-226 Warner: Chapter 6
12/7	Hirzebruch's Signature Theorem	Section 19: pp219-226 Warner: Chapter 6
12/12	Exotic Smooth Structures on S⁷	Milnor's paper
12/14	Atiyah-Singer Index Theorem	Spin Geometry: Chapter III

This page is maintained by Aleksey Zinger.
Last modified: December 2, 2006.