MATH 439-001 40631 ST: Intro to Diff Manifold

Class time  8:00 am-9:15 am Location SMLC 120.

                                                                 

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

 

Office Hours: Tuesday & Thursday 2:00pm-3:00pm. You are also welcome to stop by anytime you have a question.

 

Final exam:  May 10, 7:30am - 9:30am, double check the Final Examination Schedule.

Students having conflicts with this exam schedule must notify the appropriate instructor before Friday, March 30, 2018.      

 

Textbook: Gauge Fields, Knots and Gravity Authors: John Baez & Javier P Miranda

Topic of the course: Part I) Manifolds and Calculus on manifolds, Vector Fields, Differential Forms and DeRham Theory, Maxwell's Equations, Lie groups and Lie Algebras. Part II) Basics on Bundles and Connections, Gauge transformations, Holonomy, Curvature and the Yang-Mills Equation. Prerequisites: 264, 321. 322 will be helpful

Other Textbooks you might find useful:

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).

 

 

 

                                            All Homework assignments are posted 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. 15

 Differential of a map. Topological spaces.

HW 1 Posted on UNMLearn

Jan. 22

 Closed and compact sets. Differentiable manifolds, smooth maps, diffeomorphic and non-diffeomorphic manifolds.

HW 2 Posted on UNMLearn

Jan. 39

 Stereographic projection. Submanifolds. The rank theorem. Examples.

HW 3 Posted on UNMLearn

Feb. 5

 Submanifolds of M_n - GL(n), Sym(n), O(n), SO(n), SL(n). Lie groups. Derivations/vector fields on a manifold.

 

 HW 4 Posted on UNMLearn

Feb. 12

 Local expressions for vector fields - change of coordinates and compatibility. Local property and existence of bump functions. The tangent space.

HW 5 Posted on UNMLearn

Feb. 19

 The tangent space as derivations and velocities of curves. Expressions in local coordinates. Push forward of tangent vectors. The tangent bundle.

HW 6 Posted on UNMLearn

Feb. 26

 Flows of vector fields - (local) 1-parameter groups of diffeomorphisms, examples, the fundamental theorem on existence, uniqueness and dependance on parameters, straightening of a vector field. (Frobenius' theorem) The Lie derivative. HW 7 Posted on UNMLearn

Mar. 5

 The Lie derivative of a vector field - commutativity of the flows and the Lie derivative. Multilinear and skew-symetric forms on a vector space. Tensor product of finite dimensional vector spaces. HW8 Posted on UNMLearn

Mar. 12

Spring Break

 

Mar. 19

   Alternating and symmetric multilinear forms; k-forms, the wedge product of forms.     HW9 Posted on UNMLearn

 

Mar. 26

 

   Linear isomorphism between vector spaces and corresponding isomorphisms between the tensor algebras. Tensor fields on a manifold. Differential forms on a manifold - exterior product derivative. Derivations and skew-derivations of the exterior algebra and their properties. HW10 Posted on UNMLearn

 

Apr. 2

 Constructing new derivations or skew -derivations from given derivations and skew-derivations using the commutator and anti-commutator. The exterior derivative, the interior derivative, the Lie derivative. Cartan's formula. Closed and exact forms. Poincare's lemma. DeRham cohomology. Integrals of forms with compact support over manifolds without boundary.

 

 HW11 Posted on UNMLearn (will not be collected due to exam next week)

Apr. 9

 Integrals of forms with compact support over manifolds without boundary. Orientabilty and volume form. Stokes theorem. Manifolds with boundary. Orientation of the boundary. Midterm Exam (Thursday) HW11 Posted on UNMLearn (turn in on April 19)

Apr. 16

 Proof of Stokes theorem on a manifold with boundary. Flux (through a surface) and circulation (on a loop) of a vector field. Basic formulas of vector calculus as corollaries from Stokes' formula. Non-trivial cohomology class of degree 1 and the fundamental group. HW12 Posted on UNMLearn

Apr. 23

 Existence of non-trivial cohomology class of degree k and de Rhamm duality. Cohomology of spheres. Non-degenerate bilinear forms and the dualities they define. Poincare diality. The Hodge-*. HW13 Posted on UNMLearn

Apr. 30

 (Monday-make-up class.) The Hodge-* in Minkowski and Euclidean spaces. Geometric formulation of Maxwell's system. The potential (gauge) 1-form. Vector bundles, connection, curvature and Maxwell's system. HW14: Posted on UNMLearn (movie)

May 9 Finals week

Final Exam May 10, 7:30am - 9:30am