MATH 439-001 40631 ST: Intro to Diff Manifold

Class time  MWF 11:00 am-11:50am Location SMLC 356.

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. You are also welcome to stop by anytime you have a question.

 

Final exam:  check the Final Examination Schedule.

Students having conflicts with this exam schedule must notify the appropriate instructor before TBD

 

Textbook: Introduction to Geometry and Topology, ISBN 9783034809825/ 3034809824, Ballmann, Werner
(free on the UNM network)

Supplemental books:

  1. Baez J., Muniain J., Gauge Fields, Knots and Gravity,
  2. do Carmo M.P. Differential geometry of curves and surfaces. (free on the UNM network)
  3. Lafontaine J., An Introduction to Differential Manifolds (free on the UNM network)
  4. Hamilton M., Mathematical Gauge Theory, (free on the UNM network).

Description: The main goal of the course is to familiarize you with geometric structures through a gentle introduction to some topics motivated by physics and geometry, such as, Maxwell's Equations, Manifolds, Vector Fields, Differential Forms and DeRham Theory, Lie groups and Lie Algebras, Bundles and Connections, Curvature and the Yang-Mills Equation. Prerequisites: 264, 321. 431 will be helpful.

 

Please note the following guidelines for the course:

 

Grades: The final grade will be determined by  attendance (100 pts), homework (100 pts), one midterm exam (100 pts) and a final exam (100 points).  The Final Exam score will replace the midterm score if the latter is lower than the Final Exam score. The exams will mainly cover material from Part I.  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.   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).

 

Syllabus and Homework – Spring 2020 

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

 

 

Class Week

Topics and Section

Homework (due the 2nd class of the following week) in MS Teams
1. Jan 20 Smooth maps between Euclidean spaces. Coordinate charts, atlasses and smooth manifolds. HW
2. Jan 27 Manifold construction lemma. Smooth bump functions, smooth partition of unity. Lie groups. Tangent vectors. HW
3. Feb 3 The tangent space. HW
4. Feb 10 The tangent bundle. The rank theorem. Discussion of HW Problem 3.4. Injective immersions and embeddings. HW
5. Feb 17 Embeddings - slice theorem; level sets of maps; Lie subgroups HW
6. Feb 24 Closed Lie subgroups. Examples. Quaternions, the group Sp(1). HW
7. March 2 SU(2)=Sp(1) =Spin (3) and SO(3). Spin(4) and SO(4). The Lie bracket. HW
8. March 9 F-related vector fields. The Lie algebra of a Lie group. Free and proper Lie group actions and quotient spaces. HW Homework 8 due March 25
9. March 16 Spring Break  
10. March 23 Principle fiber bundles. Transition functions. Vector bundles. HW9
11. March 30 Vector bundles - examples, constriction, cotangent bundle, differential forms. HW10
12. Apr 6 Differential forms and tensors. Midterm Exam (take home) HW11
13. Apr 13 Derivations and contractions of tensor fields. Derivations and skew-derivations of differential forms - interior product, exterior derivative, Lie derivative. de Rham cohomology. HW12
14. Apr 20 Cohomology and homotopy. Complete vector fields and their flows, the Lie derivative. Cohomology of R^n. Orientation and volume form. Stokes' theorem on a manifold without boundary HW13
15. Apr 27 Stokes' theorem. Computing cohomology, cohomology with compact support. Semi-Riemannian metrics. HW14
16. May 4 Semi-Riemannian metrics, Hodge-*. Geometric form of Maxwell's system.  
Wednesday May 13 Final Exam 10:00am.‐12:00pm, available in Teams at 9:30am.