MATH230A: Differential Geometry
Outside Link by Selena Zhang and Aayush Gupta https://www.coursetexts.org/differential-geometry-math230a
[IMPORTANT] The syllabus is updated.
Time: 10:30-11:45 pm, Tuesday-Thursday
Place: Science Center 221
Instructor: Fan Ye, Science Center 505H, fanye@math.harvard.edu
Course Assistant: Keeley Hoek, Science Center 333G, khoek@math.harvard.edu
Office Hours:
Fan Ye, SC 505H, Tuesday 1:30-3:20 pm.
Keeley Hoek, SC 333G, Thursday 2:00-3:00 pm
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_…
- 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. 1-8): smooth manifolds and Lie groups [Taubes, Chapters 1-2]
- Weeks 3-4 (Sept. 13-22): vector bundles [Taubes, Chapters 3-6]
- Weeks 5-6 (Sept. 27-Oct. 6): metrics and geodesics [Taubes, Chapters 7-9]
- Weeks 7-8 (Oct. 11-20): de Rham cohomology and covariant derivative [Taubes, Chapters 12]
- Weeks 9-10 (Oct. 25-Nov. 3): Levi-Civita connections and Riemann curvature tensors [Taubes, Chapters 15, 14.1]
- Weeks 11-13 (Nov. 8-22): characteristic classes and principal bundles [Taubes, Chapters 14, 10-11]
- Week 14 (Nov. 29-Dec. 1): Yang-Mills equation and applications.