Math 208 Manifolds I Fall 2010

Updated 10/4/10

INSTRUCTOR

Instructor: Debra Lewis
Office: 359B Baskin Engineering
Phone: 459-2718
E-mail: lewis at ucsc dot edu (checked more often than voicemail or gmail) and/or DebraKLewis at gmail dot com

TIMES AND PLACES

Office hours: Tuesday 2:30-3:30, Wednesday 2:30-3:30, Friday 12:00-1:00
Course web page: http://people.ucsc.edu/~lewis/Math208 (here)

TEXT

Introduction to Smooth Manifolds, John M. Lee, Springer.

Possible supplemental texts: An Introduction to Differentiable Manifolds and Riemannian Geometry, Boothby. Topology from a Diffential Viewpoint, Milnor.

Some material on the phase flow and matrix exponential:

My old lecture notes on the phase flow and matrix exponential
Mathematica notebook with sample plots of flows of the unit square. Available as actual notebook or PDF firmcopy. (current version: 10/24/07)
Scholarpedia Article on Dynamical Systems by James Meiss (dynamical systems big fish) of the University of Colorado. The section on Flows is the relevant part, but the whole article is nice.
Wikipedia This actually looks like a reasonably good concise overview of the matrix exponential.

Some material on the Jordan Normal Form of a matrix:

My old lecture notes on the Jordan normal form
Mathematica notebook with sample 4x4 Jordan normal form and matrix exponention calculations. Available as actual notebook or PDF firmcopy. (current version: 10/24/07)
MIT OpenCourseWare This looks like a good, fairly sophisticated treatment.
Kahan's course notes William Kahan, of UC Berkeley, is one of the superstars of numerical analysis and King of Rigor. Lots of good information, beautifully presented, but some of the material is probably a bit advanced for our purposes.
Wikipedia The first few sections are relevant, but possibly too cryptic to be very useful.

The midterm. (11/5/08)

Topics:
Definition of manifolds, tangent bundle, inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem, vector fields, flows, and Lie bracket, Frobenius' theorem.

VERY TENTATIVE SCHEDULE

December 2: Additional topics and/or presentations

Monday	Wednesday
September 27: Topological and smooth manifolds	September 29: Smooth functions and maps
October 4: Tangent vectors and derivations	October 6: Pushforwards, coordinate calculations, and partitions of unity
October 11: Submersions, immersions, and embeddings	October 13: Inverse and Implicit Function Theorems
October 18: IFT continued and submanifolds	October 20: Level sets and Lie groups
October 25: Vector fields and the tangent bundle	October 27: Integral curves and flows
November 1: Fundamental flow theorem, existence and uniqueness	November 3: Lie brackets
November 8: Distributions and involutivity	November 10: Frobenius' Theorem
November 15: Singular points, Sard's Theorem	November 17: Whitney Embedding Theorem
November 22: Whitney Approximation Theorem and tubular neighborhoods	November 24: Homotopies and approximations
December 4: Additional topics and/or presentations

COURSE WORK

Weekly homework assignments. Most exercises will be from the text, but some may be taken from other sources (these will be provided as hard copies and online).
Take-home final.
Optional presentation.