MATH 539-001 Selected Topics Geometry & Topology,

Time TTh 1400-1515, Location SMLC 352.


Instructor: Dimiter Vassilev     Office :  SMLC, Office 326  Email: 

Phone Number: 505 277 2136


Office Hours: Monday 3:30-4:30pm, Wednesday 4pm-5pm, Thursday 4:30-5:30pm. Feel free to stop-by anytime when you have a quick question.


Description: Continuation of MATH 537 and MATH 538: comparison results in Riemannian and sub-Riemannian geometry (Sasakian and contact geometry), variational problems in geometry and geometric flows. Prerequisite: 537 & 538 or instructor permission.

Textbooks: Class notes.

HOMEWORK: Homework and course related material will be posted on UNMLearn. The general rule is that you should do the homework problems in order to maximize what you learn in the class.

ATTENDANCE: Attendance at UNM is mandatory, see policy.

Schedule and homework per week

  1. Aug 20: Review. Computations in normal coordinates. Homework
  2. Aug 27: Computations in normal coordinates. Integration on a Riemannian manifold. Volume of small balls (and the scalar curvature). Homework
  3. Sep 3: Curvature and volume - Bishop's theorem Homework
  4. Sep 10: Gunter's theorem. Rauch's comparison theorem. Homework
  5. Sep 17: The cut-locus. Homework
  6. Sep 24: More on the cut-locus. The distance function. Homework
  7. Oct 1: Hessian and Laplacian comparison theorems. Homework
  8. Oct 8 Closed geodesics and the cut locus and estimates on the injectivity radius- Klingenberg's theorem. The sphere theorem. Homework (October 11-12, Fall Break):
  9. Oct 15: Finish the proof of Klingenberg's theorem in even dimensions, the Bishop-Gromov volume comparison theorem, S-Y Chang's maximum diameter theorem. Toponogov's comparison theorem. Homework
  10. Oct 22: Applications of Toponogov's comparison theorem. Homework
  11. Oct 29: The Gauss-Bonnet-Chern formula. Homework
  12. Nov 5: Continue the Gauss-Bonnet-Chern formula, the Poincare-Hopf index theorem. Minimal surfaces - 1st variation formula. Homework
  13. Nov 12: Parametric minimal surfaces. Calibrations. Bernstein's theorem. The Monotonicity theorem. Homework
  14. Nov 19 Proof of the monotonicity theorem. Homework (November 22-25 Thanksgiving Break):
  15. Nov 26: The Bernstein's theorem for a minimal graph. The Simons' minimal cone. Second variation and stable minimal hypersurfaces - Simons' identity. Homework
  16. Dec 3: The general Weitzenbock formula