Math 225A: Differentiable Manifolds
Fall 2019
Lectures:
MWF 2pm - 2:50pm,
Location: MS 6201
Discussion: Tu 2pm - 2:50pm, Location: MS 5233
Syllabus
This is the first quarter of a year-long
sequence in geometry and topology.
Instructor: Ko Honda
Office: MS 7919
Office Hours: Mon 1-1:50pm, Wed 11am-noon
E-mail: honda at math dot ucla dot
edu
URL: http://www.math.ucla.edu/~honda
TA: Austin Christian; office
hours TBA; archristian at math dot ucla dot edu
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.
- Differential forms: tensor and
exterior algebra, exterior differentiation, and Lie
derivatives.
- Integration: Stokes' theorem, de
Rham cohomology, and computations using Meyer-Vietoris
sequences.
- Vector fields, distributions,
Frobenius' 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.
Homework
There will be weekly problem sets; see
the class
schedule. Homework is due on Mondays,
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
I will follow my Differential Geometry Course
Notes. The main reference is Lee's book, where you
can find more details and examples.
- Differential
Geometry Course Notes
- Lee, Introduction to Smooth Manifolds
- Spivak, A Comprehensive
Introduction to Differential Geometry
- Tu, An Introduction
to Manifolds
- Warner, Foundations of Differentiable Manifolds and
Lie Groups
- Peter
Petersen's notes
WARNING: The course syllabus provides a general
plan for the course; deviations may become
necessary.
Last modified: September 26,
2019. |