MATH 439001 40631 ST: Intro to Diff Manifold
Class time 8:00 am9:15 am Location SMLC 120.
Instructor:
Dimiter Vassilev Office :
SMLC, Office 326
Email: vassilev@unm.edu
Phone
Number:
505 277 2136
Textbook: Gauge Fields, Knots and Gravity Authors: John Baez & Javuer 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 YangMills Equation. Prerequisites: 264, 321. 322 will be helpful
Other Textbooks you might find useful:
http://link.springer.com/book/10.1007%2F9783319207353 (available free for download on the UNM network)
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:
Makeup 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 2773506. In addition, they should see CATS
Counseling and Therapy Services;
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 nondiffeomorphic 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) 1parameter 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 skewsymetric forms on a vector space. Tensor product of finite dimensional vector spaces.  HW8 Posted on UNMLearn 
Mar. 12 
Spring Break 

Mar. 19 

HW9 Posted on UNMLearn

Mar. 26 
HW10 Posted on UNMLearn


Apr. 2 

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. Nontrivial cohomology class of degree 1 and the fundamental group.  HW12 Posted on UNMLearn 
Apr. 23 
Existence of nontrivial cohomology class of degree k and de Rhamm duality. Cohomology of spheres. Nondegenerate bilinear forms and the dualities they define. Poincare diality. The Hodge*.  HW13 Posted on UNMLearn 
Apr. 30 
(Mondaymakeup class.) The Hodge* in Minkowski and Euclidean spaces. Geometric formulation of Maxwell's system. The potential (gauge) 1form. 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 
