Math 225B: Differential Geometry

Winter 2018

Instructor: Sucharit Sarkar.
Class: MWF 1-1:50pm GEOLOGY 6704.
Office hours: M 2-4pm MS 6909.

TA: Sangjin Lee.
TA review session: Th 1-1:50pm GEOLOGY 6704.
TA office hours: Tu 4-6pm MS 3919.

Textbook: Additional reading: Useful links: University page, Department page, MyUCLA, old Qualifying exams (QE).

Topics: 225B is a course in differential geometry. The students are required to know the material of 131AB (real analysis), 120A (basic differential geometry), 121 (point set topology), and 225A (differential topology). This is intended to be a course for the first year graduate students to help them prepare for the Topology Qualifiers, but advanced Math majors with sufficient background are allowed to take the course as well. The course covers an assortment of topics such as Lie derivatives, integrable distributions and Frobenius theorem, differential forms, integration and Stokes theorem, de Rham cohomology, including Mayer-Vietoris sequence, Poincare duality, Thom classes, degree theory and Euler characteristic revisited from viewpoint of de Rham cohomology, Riemannian metrics, gradients, volume forms, and interpretation of classical integral theorems as aspects of Stokes theorem for differential forms.

Homework: Homeworks are due at the beginning of lecture on Wednesdays. Do not submit homework by e-mail. No late homework will be accepted.
HW Due on Problems
1 1/17 HW1
2 1/24 HW2
3 1/31 HW3
4 2/7 HW4
5 2/14 HW5
6 2/21 HW6
7 2/28 HW7
8 3/7 HW8
9 3/14 HW9
You are encouraged to work in groups on your homework; this is generally beneficial to your understanding and helps you learn how to communicate clearly about mathematics. However, you must write up all solutions yourself. Moreover, since crediting your collaborators is an important element of academic ethics, you should write down with whom you worked at the top of each assignment. You should also cite any sources (other than lectures and the textbook) that you use.

Exams: There is a single final exam which is non-collaborative and closed-book. You are not allowed to use books, notes, or any electronic devices (such as calculators, phones, computers) during the exam.
Location, Date, Time: Tu 3/20 11:30am-2:30pm GEOLOGY 6704.
There will be no make-up exams. Attending the final exam is mandatory. In particular, note that university policy requires that a student who misses the finals be automatically given F, unless the absence is due to extreme and documented circumstances, in which case, if the performance in the course is otherwise satisfactory, the grade might be I.

Grading: Numerical grades will be recorded in the MyUCLA gradebook and the composite numerical grade will be computed as 70% HW + 30% Final, and the final letter grade will be assigned based on that.
If you believe a problem on a homework or an exam has been graded incorrectly, or that your score was not correctly recorded in the MyUCLA gradebook, you must bring this to the attention of the instructor within 10 calendar days of the due date of the assignment in question, or the date of the exam, and before the end of the quarter (3/23). Grading complaints not initiated within this period of time will not be considered. Please verify in a timely manner that your scores are correctly recorded on MyUCLA.

Tentative schedule:
Week Lectures Dates Topic
1 3 1/8-1/12 S 2-3, smooth manifolds, tangent bundles, vector fields
2 2 1/17-1/19 S 3-4, vector bundles, orientations, duals, tensor products
3 3 1/22-1/26 S 5, flows along vector fields, Lie derivatives
4 3 1/29-2/2 S 6, Frobenius integrability theorem, local and global
5 3 2/5-2/9 S 7, differential forms, deRahm cochain complex
6 3 2/12-2/16 S 7, Cartan's magic formula, Poincare lemma
7 2 2/21-2/23 S 11, Mayer Vietoris sequence, cohomology of S^n
8 3 2/26-3/2 S 8, Stokes theorem, cohomology with compact support
9 3 3/5-3/9 S 11, Poincare duality, degree, deRahm theorem
10 3 3/12-3/16 S 11, Thom class, Euler class, index of vector fields