MATH 539-001 Selected Topics Geometry & Topology,

Time MWF 11:00-11:50, Location: SMLC 120.

                                                                 

Instructor: Dimiter Vassilev     Office :  SMLC, Office 326  Email: vassilev@unm.edu 

Phone Number: 505 277 2136

 

Office Hours: Monday & Friday 15:00-16:00. Feel free to stop-by anytime when you have a quick question.

 

Description: Sub-Riemannian geometry: CR, Sasakian and quaternionic contact geometry, variational problems in geometry and geometric flows, comparison results in sub-Riemannian geometry. The goal is to coverrecent topics and intorduce open problems in the subjects. Prerequisite: instructor permission.

Texts: I will not follow any particular text overall. Your class notes and published papers (posted on UNMLearn) will be your main source of information. However, here are some books, which offer a more expanded version of some of the topics, in addition to many other topics.

• Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications
• Agrachev, Barilari, Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry
• Barilari, Boscain, Sigalotti, eds., Geometry, Analysis and Dynamics on Sub-riemannian Manifolds Volume I
• Tanaka, A differential geometric study on strongly pseudo-convex manifolds
• Capogna, Danielli, Pauls, Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem
• Biquard, Asymptotically Symmetric Einstein Metrics
• Biquard, ed., AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries
• Boyer, Galicki, Sasakian Geometry
• Ivanov, Minchev, Vassilev, Quaternionic Contact Einstein Structures and the Quaternionic Contact Yamabe Problem
• Ivanov, Vassilev, Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem
• Dragomir, Tomassini, Differential Geometry and Analysis on CR Manifolds

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 19: Sub-Riemannian structures - Hormander's condition; examples -The Siegal upper half space and its boundary. Homework
2. Aug 26: Pseudohermitian structure on the Heisenberg group. Geodesics. Isoperimetric inequality. Read Article
3. Sep 2 (labor day): Symplectic manifolds. Normal and singular geodesics. Reading.
4. Sep 9: Riemannian approximation. Gromov-Hausdorff convergence. Article
5. Sep 16: Gromov-Hausdorff convergence. Length metric spaces. Article
6. Sep 23: Sub-Riemannian spaces as length metric spaces. Completeness.
7. Sep. 30: The Riemannian approximations and their Gromov-Hausdorff convergence to the sub-Riemannian metric on the Heisenberg group. Homogeneous groups. Carnot groups. Article
8. Oct 7: Properties of the norm. Haar measure existence and uniqueness. (October 10-13, Fall Break):
9. Oct 14: Unimodular groups. Haar measure on homogeneous groups. Hausdorff measure.
10. Oct 21: Hausdorf dimension. Ahlfors regulararity. The ball-box theorem. Homogeneous dimension and Ahlfors regular measures on a sub-Riemannian manifold
11. Oct 28: Linearly adapted coordinates, priviledged coordinates. Examples - article. The tangent space as a homogeneous nilpotent group - reading, reading for the following weeks.
12. Nov 4: More on priviledged coordinates. Examples.
13. Nov 11: Distance estimates. Asymptotic cones and tangent spaces.
14. Nov 18: Approimate isometries and Gromov-Hausdorff convergence. Homogeneous spaces. Proof of the Bellaiche-Mitchell's theorem.
15. Nov 25: Characteristics. (November 28 Thanksgiving Break)
16. Dec 2: Pontryagin's maximum principle, the endpoint map and the charactersitics.