Math 225B:  Differentiable Manifolds

Winter 2020

Lectures: MWF 2pm - 2:50pm,  Location: MS 5138
Discussion: Th 2pm - 2:50pm, Location: MS 5138

Syllabus

This is the second quarter of a year-long sequence in geometry and topology.

Instructor: Ko Honda
Office: MS 7919
Office Hours: Wed 11am-noon, 1-1:50pm
E-mail: honda at math dot ucla dot edu
URL: http://www.math.ucla.edu/~honda

TA: Eilon Reisin-Tzur; office hours TBA; ereisint at math dot ucla dot edu

Topics

  1. Sard's theorem and transversality.
  2. Oriented intersection theory, degree, Lefschetz fixed point theorem.
  3. Poincaré duality, Thom isomorphism, Pontryagin-Thom theory
  4. Hodge theory, elliptic operators

Prerequisites

  • Knowledge of basic manifold theory (e.g., Math 225A)
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

For the differential topology portion of the course:
  1. Guillemin & Pollack, Differential Topology,
  2. Milnor, Topology from the Differentiable Viewpoint.
For Poincaré duality and the Thom isomorphism:
  1. Bott & Tu, Differential Forms in Algebraic Topology.
For the Hodge theory portion of the course:
  1. Differential Geometry Course Notes, second semester,
  2. Warner, Foundations of Differentiable Manifolds and Lie Groups.
 
WARNING:  The course syllabus provides a general plan for the course; deviations may become necessary. 

Last modified: January 2, 2020.