Web page: http://www.math.ucla.edu/~cm/225a.1.12f/index.html
Prerequisites: Real analysis in several variables (e.g. the implicit function theorem) and point set topology.
Topics to be covered: Manifolds, tangent vectors, smooth maps, tangent bundles and vector bundles in general, Sard's theorem on the measure of critical values, embedding theorems, vector fields and integral curves, Ehresmann's fibration theorem, transversality, degree theory, Lefshetz fixed-point theorem, Euler characteristic.
Grading: 50% homework, 50% in-class final.
Homework: Homework will be assigned every week and will be due the following Friday. The homework assignments will be handed out in class and will also be posted on the web page. You must hand in the homework in class each Friday. You are encouraged to talk about the problems with other students, but you should write up the solutions individually. You should acknowledge the assistance of any book, student or professor. The lowest homework score will be dropped.
Problem Set 1 (due October 5)
Problem Set 2 (due October 12)
Problem Set 3 (due October 19)
Problem Set 4 (due October 26)
Problem Set 5 (due November 2)
Problem Set 6 (due November 9)
Problem Set 7 (due November 16)
Problem Set 8 (due Monday, November 26)
Problem Set 9 (due December 7)
Final Exam: The final will take place in class, on Friday,
December 14, from 11:30am to 2:30pm.
Here is a practice exam and also an old exam.
Office hours (Prof. Manolescu) for the week of the final: Monday 11-12, Wednesday 2-3, Thursday 11-12.
The TA will run a review section on Tuesday December 11, 6-8pm, in MS 5147.