Math 225A: Differential Topology
Lectures:
MWF 11-11:50am,
Location: MS 5148
Discussion: Th 11-11:50am, Location: MS 6221
Syllabus
This is the first quarter of a year-long
sequence in geometry and topology.
Instructor: Ko Honda
Office: MS 7901
Office Hours: Wed. 1-3pm
E-mail: honda at math dot ucla dot
edu.
Telephone: 310-825-2143
URL: http://www.math.ucla.edu/~honda
TA: Michael Menke
Topics
- Review of advanced calculus
(calculus on R^n); inverse and implicit function
theorems.
- Differentiable manifolds and their
maps.
- Tangent and cotangent bundles, vector
bundles.
- Vector fields and integral curves.
- Sard's theorem, transversality,
Whitney embedding theorem.
- Oriented intersection theory: degree,
Lefschetz fixed point theorem.
Prerequisites
- Knowledge of calculus on R^n, as
presented in the first three chapters of Spivak's Calculus on Manifolds
book.
- This course requires more mathematical
maturity than the average first-year graduate course
in the mathematics department. I will not follow
any text closely in the first half of the course, and
it is up to you to look up the necessary information
from the various references. (Of course, I would
be happy to suggest where to look up things.)
Homework
There will be weekly problem sets; see
the class
schedule. Homework is due on Fridays,
although there may be some exceptional weeks. The
problem sets count for a large percentage of your total
grade (approximately 70%). You may work with
others or consult other textbooks, but the homework you
turn in must be written by
you, in your own words, and you must cite your
sources used and your collaborators!
Final
examination
There will be a
take-home final. This will be approximately 30% of
your final grade.
References
In the first half of the course I will follow my
Differential Geometry Course Notes and in the second half
I will loosely follow Guillemin & Pollack.
- Differential
Geometry Course Notes
- Guillemin & Pollack, Differential
Topology.
- Morita, Geometry of Differential Forms.
- Warner, Foundations of Differentiable Manifolds and
Lie Groups.
- Boothby, An
Introduction to Differentiable Manifolds and
Riemannian Geometry
- Peter
Petersen's notes
WARNING: The course syllabus provides a general
plan for the course; deviations may become
necessary.
Last modified: September 26,
2014. |