MATH 439-001 55949 ST: Intro to Diff Manifold & Introduction to Differentiable Manifolds - 51950 - MATH 536 - 001

Class time  11:00 am-12:15 pm Location SMLC 352.

                                                                 

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

 

Office Hours: Wednesday 1:30pm-2:30pm, Tuesday & Thursday 2-3pm. You are also welcome to stop by anytime.

 

Final exam:  May 10, 12:30pm - 2:30pm, double check the Final Examination Schedule.

Students having conflicts with this examination schedule must notify the appropriate instructor before April 5, 2016. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before April 5, 2016, the instructor shall make arrangements to give a special examination. Conflicts arising as a result of scheduling out of normal hours-pattern or day sequences must be resolved by the instructor of the off-pattern courses. Changes in this examination schedule are not permitted except by formal approval of the instructor’s College Dean.       

 

Required textbook: An Introduction to Differential Manifolds (2015), Authors: Jacques Lafontaine

http://link.springer.com/book/10.1007%2F978-3-319-20735-3 (available free for download on the UNM network)

Topic of the course: Chapters 1-6 of Lafontaine’s book. Description: Concept of a manifold, differential structures, tangent and cotangent bundles, embedding, immersions and submersions, transversality, differential forms and integration, Stokes' theorem, Lie groups. 

Other Textbooks you might find useful:

Please note the following guidelines for the course:

Prerequisites: Linear Algebra (MATH321), Calculus III (MATH264) and at least two of the following courses: Topology (MATH431), Analysis (MATH401) , Analysis (MATH402).

Grades: The final grade will be determined by the homework (100 pts), two midterm exams (100 pts) and a final exam (200 points).  All grades will be posted on UNM Learn.

 

Homework:  You can work together on the homework, but you do need to write up 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 5 problems (4 pts each) for a total of 20 points.  The Extra Credit Problems are not due and are truly extra credit (8 pts each). It is best if you work and discuss the problems together and with me as questions arise. The highest 10 homeworks will be counted.

 

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).    

 

 

 

                                             Homework is hosted on UNM Learn

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

Week of

Topics Covered

Homework due Thursday following week at the beginning of class.

Jan. 18

1. Differentials, rank theorem - diffeomorphisms.

HW1 due Jan. 28

Jan. 25

2. The rank theorem,  immersions, submersions. Smooth submanifolds of Rn.

HW2 due Feb. 4

Feb. 1

3. Level sets, graphs and parametrizations. Examples.

HW3 due Feb. 11

Feb. 8

4. The tangent space. Sets of measure zero and their images under smooth maps. Critical points and values.

 

HW4 due Feb. 18

Feb. 15

5. Sard's theorem. Transversality. Topological and smooth manifolds.

HW5 due Feb. 25

Feb. 22

6. Construction of Manifolds. Smooth Maps - non-equivalent smooth structures, diffeomorphic smooth structures. Examples -  Projective Spaces.

HW6 due Mar. 3

Feb. 29

7. Fibrations, the Hopf fibration. The Fundamental Theorem of Algebra.

HW7 due Mar. 10,

Feb. 29 Homework Session SMLC 352 9:45-10:45

Mar. 7

8. Tangent space. Local diffeomorphisms, immersion, submersion, submanifolds.

Midterm Exam 1

HW8 due Mar. 31

Mar. 14

Spring Break

 

Mar. 21

 

 

 

 

 

Mar. 28

 

 9.  Derivations and vector fields. Tangent bundle. Vector bundles.

 

HW9 due Apr. 7

make-up class SMLC 352 9:30-10:45

Apr. 4

 10. Smooth Partition of unity. Whitney embedding Theorems. Flows of vector fields,.

 

 

HW10 due Apr. 14

Apr. 4 Homework Session SMLC 352 9:45-10:45

Apr. 8 make-up class SMLC 352, 9:30-10:45

Apr. 11

11.  The Lie bracket. The Frobenius theorem. Multilinear Algebra.

 

HW11 due Apr. 21

Apr. 18

12. The cotangent bundle. Differential forms - derivations (exterior derivative, Lie derivative, interior product, Cartan's formula).

Midterm Exam 2

 

HW12 due Apr. 28

Monday Homework Session SMLC 352 9:45-10:45.

Make-up classes: Wednesday & Friday 9:45am-10:55am.

Apr. 25

13. Integration of differential forms. Manifolds with boundary. Stokes theorem. DeRham cohomology. Degree of a map.

 

14. Lie groups - left invariant vector fields, the Lie algebra, exponential mapping, Lie subgroups. Group actions. Quotient manifolds and homogeneous spaces.

HW13 due May 5

Make-up class: Monday 9:45am-10:45am

No class on Tuesday

Make-up class: Wednesday 9:45am-10:45am

Make-up class: Friday 9:00am-10:15am

May 2

 

No classes, but turn in the due homework on Thursday as usual and collect your old homework.

May 9 Finals week

Final Exam May 10, 12:30pm - 2:30pm