CRN 46637 Math 536 001 Intro Diff Manifolds

Lecture Time 9:00-9:50, SMLC 352

NOTE: class and office hours will be through Microsoft Office Teams at the usual times.

                                                                

Instructor: Dimiter Vassilev     Office :  SMLC, Office 326  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Office Hours: Monday & Friday 2pm - 3pm. Feel free to stop-by anytime when you have a quick question.

 

Final exam: check the official Final Examination Schedule.

Students having conflicts with this exam schedule must notify the appropriate instructor before TBD. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before TBD, the instructor shall make arrangements to give a special examination. Changes in this examination schedule are not permitted except by formal approval of the instructor’s College Dean.  

 

Textbook:

 

Other suggested textbook:

Please note the following guidelines for the course:

 

Description: Concept of a manifold, differential structures, vector bundles, tangent and cotangent bundles, embedding, immersions and submersions, transversality, Stokes' theorem.

 

Prerequisite: Math 511.


Grades: The final grade will be determined by homework (100 points), one midterm exam (100 pts) and a final exam (200 points). Not all homework problems will be graded. You will get credit if you turn in the statements of all the homework problems written in the correct order. The Final Exam score will replace the midterm score if the latter is lower than the Final Exam score.  All grades will be posted on UNM Learn.

 

Homework:  You can work together on the homework, but you do need to write your own solutions in your own words. To help the grader, please write your solutions up neatly and clearly and staple the sheets.   Each homework has a total of 20 points.  The Extra Problems (if any) are not due. Not all homework problems will be graded.

 

Missed Exams:  Make-up exams can be arranged for exams missed with a VALID excuse (illness, family emergency, active participation in scholarly or athletic activities), and ONLY if prior notice is given. 

 

Disability Statement: We will accommodate students with documented disabilities. During the first two weeks of the semester, those students should inform the instructor of their particular needs and they should also contact Accessibility Services in Mesa Vista Hall, Room 2021, phone 277-3506.  In addition, they should see CATS- Counseling and Therapy Services; Student Health Center (277-4537). (They can help if you suffer from exam anxiety).    

 

Syllabus and Homework – Spring 2020 

(please check after class as advanced postings of homework could change)

 

 

Class Week

Topics and Section

(Galen's course Notes)

Homework (due the 2nd class of the following week) in MS Teams
1. Jan 20

 

 1. Topological and smooth manifolds - definitions and examples, paracompactness, continuous partition of unity, smooth structures defined by an atlas. HW1
2. Jan 27 Examples using the manifold construction Lemma. Smooth partition of unity. Examples of diffeomorphsims; complex structure on 2-sphere and complex projective line. HW2
3. Feb 3 Manifolds with boundary. The tangent space. HW3
4. Feb 10 The tangent bundle. The rank theorem; immersions and submersions; smooth covering maps HW4
5. Feb 17 Proper maps and covering maps. Immersed and (embedded) submanifolds HW5
6. Feb 24 Examples of sub-manifolds. Sard's theorem. Lie subgroups. Tangent space to a sub-manifold. Whitney's embedding theorem. HW6
7. March 2 Transversality. Locally trivial fiber bundles. Group actions. HW7
8. March 9 The quotient manifold theorem. Principle fiber bundles.

(Galen's Notes)

 HW Homework 8 due March 25
9. March 16 Spring Break  
10. March 23 Free and proper actions and PFB. Vector bundles. HW9
11. March 30 Vector bundels - constructions. Bundle homomorphisms and sections. HW10
12. Apr 6

Tensors. Differential forms. Characterization of tensor fields. Pull-backs of covariant tensors; diffeomorphsims and tensor fields.

Midterm Exam (take home)

 HW11
13. Apr 13 Derivations and contractions of tensor fields. The Lie derivative. Derivations and skew-derivations of differential forms - interior product, exterior derivative, Lie derivative. HW12
14. Apr 20 de Rham cohomology, Poincare's lemma. Induced map on cohomologies, properties under homotopies. Cohomology and homotopic spaces. Volume form and Orientation. Integration of forms. Stokes' theorem. HW13
15. Apr 27 The general Stokes' theorem. Computing cohomology - Mayer-Vietoris sequence, cohomology of spheres. Poincare lemma and cohomology with compact support. HW14
16. May 4 Flows of vector fields, complete vector fields - vector fields with compact suport, left-invariant vector fields. Exponential of a Lie group, differential at the identity. Integrable distributions, Frobenius' theorem.  
Wednesday May 13 Final Exam 7:30am ‐ 9:30am, available in Teams at 7am.