A minimal surface with some boundary conditions on a curve.

This is a graduate course on differential geometry, with a focus on Riemannian geometry and geometric analysis. It covers basic concepts in Riemannian geometry (Riemannian metrics, isometries, connections and geodesics), along with advanced topics including

• minimal surface theory.
• metrics of positive curvature and Ricci flow.
• systolic geometry and Gromov's systolic inequality.
• convex integration and Nash's embedding theorem.
• basic of symplectic manifolds.
• basic of (almost-)complex manifolds.
• convex integration and Nash's embedding theorem.
• Kahler geometry and the Calabi-Yau theorem.

Students should have a strong background in real analysis (525a), differential topology (535a) and algebraic topology (540). The basics of PDEs (555a) and functional analysis is also helpful.

• Instructor: Julian Chaidez (julian.chaidez@usc.edu)
• Lectures: MWF 10-11 in KAP 427
• Office Hours: MWF 11-12 in KAP 245
• Grade Scheme: 40% HW + 30% oral midterm + 30% oral final.
• Oral Exams: An oral midterm and an oral final will be administered in order to evaluate understanding. Scheduling information will be forthcoming.
• Homework: A small number of homework assignments will be assigned to assess comprehension. Homework details will be provided by email and via the schedule below.
• For more information, see the course syllabus.

Homework assignmemts will be posted here later in the semester.

• W 9/3. the three geometries, geometric functionals, calculus of variations.
• F 9/5. Riemannian manifolds, isometries, existence, distance. G+ 52-58
• M 9/8. volume, sub-manifolds, products, examples. G+ 58-65
• W 9/10. principal bundles, associated bundles, examples
• F 9/12. affine connections, principal bundle connections, Levi-Civita. G+ 70-75
• M 9/15. parallel transport, curvature of connections. G+ 75-80
• W 9/17. Riemannian, Ricci and scalar curvatures, examples. G+ 129-135
• F 9/19. geodesics, local existence/uniqueness. G+ 80-89
• M 9/22. exponential map, completeness, Hopf-Rinow. G+ 96-100
• W 9/24. first and second variation, Jacobi vector fields, applications. G+ 141-148
• F 9/26. geodesic flow, dynamics, applications. G+ 109-115
• M 9/29. area functional, variations, maximum principle. M 1-6
• W 10/1. calibration, second variational formula. M 6-12
• F 10/3. monotonicity, Bernstein Theorem, stability condition. M 12-20
• M 10/6. stability condition. M 20-29\\
• W 10/8. Simon's equation, Schoen-Simons-Yau Theorem. M 29-33
• M 10/13. Schoen-Simons-Yau Theorem, current developments. M 33-38
• W 10/15. Einstein metrics, Einstein-Hilbert functional, variations, Ricci flow. B 1-5
• F 10/17. short time existence, de Turck trick, uniqueness of Ricci flow. B 5-12
• M 10/20. evolution of length, distance. Uhlenbeck trick. B 12-17
• W 10/22. evolution of curvatures, scalar maximum principle. B 17-24
• F 10/24. applications of maximum principle, viscocity solutions. B 24-30
• M 10/27. vector bundle maximum principle. B 30-37
• W 10/29. applications, Hamilton's Ricci positive theorem. B 37-41
• Th 10/30. Oral Midterm