MATH230A: Differential Geometry

Semester: 

Fall

Offered: 

2022

Time: 10:30-11:45 pm, Tuesday-Thursday

Place: Science Center 221

Instructor: Fan Ye, Science Center 505H, fanye@math.harvard.edu

Course Assistant: Clair Dai, Science Center 431H, xdai@math.harvard.edu

Office Hours:

Fan Ye, SC 505H, Tuesday 1:30-3:20 pm.

Clair Dai, Zoom, Sunday 10:30-11:30 am (see the syllabus).

 

Main References: Differential Geometry: Bundles, Connections, Metrics and Curvature, Chapters 1-16, by Clifford H. Taubes

Prerequisites: An undergraduate-level understanding of manifolds. An undergraduate may look at the first chapter of Cliff's book for preparation.

Tentative Course Outline:

  • 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 7-9 (Oct. 11-25): de Rham cohomology, Levi-Civita connections, and Riemann curvature tensors [Taubes, Chapters 12,15-16]
  • Weeks 9-11 (Oct. 27-Nov. 10): principal bundles and connections [Taubes, Chapters 10-12]
  • Weeks 12-13 (Nov. 15-22): flat connections and characteristic classes [Taubes, Chapters 13-14]
  • Week 14 (Nov. 29-Dec. 1): Yang-Mills equation and applications.
syllabus_math230a0830.pdf67 KB
assignment_1_updated.pdf121 KB
math230a_notes_week1-2.pdf24.27 MB
assignment_2_due_oct_6_updated.pdf123 KB
math_230a_notes_week3-4.pdf25.63 MB
math_230a_notes_sept_27.pdf7.43 MB
math_230a_notes_sept_29.pdf6.24 MB
assignment_3_due_oct_20.pdf125 KB