Math230A: Differential Geometry 2023
Outside Link by Selena Zhang and Aayush Gupta https://www.coursetexts.org/differential-geometry-math230a
[Syllabus updated]
Time: 3:00-4:15 pm, Monday-Wednesday
Place: Science Center 113
Instructor: Fan Ye, Science Center 505H, fanye@math.harvard.edu
Course Assistant: Keeley Hoek, khoek@math.harvard.edu
Office Hours:
Fan Ye, SC 505H, 4:20-5:20 pm, M-W.
Keeley Hoek, SC 333G 1:30-2:30pm Tuesday.
Main References: Differential Geometry: Bundles, Connections, Metrics and Curvature, Chapters 1-16, by Clifford H. Taubes
Other suggested references:
- An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby: https://books.google.com/books/about/An_Introduction_to_Differentiable_M...
- Fundations of differentiable manifolds and Lie groups, Frank W. Warner: https://link.springer.com/book/10.1007/978-1-4757-1799-0
- Differential forms in algebraic geometry, Raoul Bott and Loring W. Tu: https://link.springer.com/book/10.1007/978-1-4757-3951-0
- Vector bundles & K-theory, Allen Hatcher https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html
Prerequisites: An undergraduate-level understanding of manifolds. An undergraduate may look at the first chapter of Cliff's book for preparation.
Tentative Course Outline: Notes will be posted on the course page after the classes.
- Weeks 1-2 (Sept. 6-13): smooth manifolds and Lie groups [Taubes, Chapters 1-2]
- Weeks 3-4 (Sept. 18-27): vector bundles [Taubes, Chapters 3-6]
- Weeks 5-6 (Oct. 2-11, with holiday on Oct. 9): metrics and geodesics [Taubes, Chapters 7-9]
- Weeks 7-8 (Oct. 16-25): de Rham cohomology and covariant derivative [Taubes, Chapters 12]
- Weeks 9-10 (Oct. 30-Nov. 8): Levi-Civita connections and Riemann curvature tensors [Taubes, Chapters 15, 14.1]
- Weeks 11-13 (Nov. 13-20, with holidays on Nov. 22-26): characteristic classes and principal bundles [Taubes, Chapters 14, 10-11]
- Week 14 (Nov. 27-Dec. 4): Yang-Mills equation and applications.