Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Course Overview

This course is meant to bring graduate students who will be using ideas from differential topology and differential geometry up to speed on these topics. Below is list of some of the highlights of the first semester.

1. Banach Manifolds and the inverse and implicit function theorems.
   
   
2. Sard's theorem (for smooth maps). Parametric transversality and some applications.
   
   
3. Whitney's embedding theorems both easy and hard.
   
   
4. Pontryagin-Thom construction. Fruedenthal suspension theorem. Computation of the low dimensional stable homotopy groups of spheres.
   
   
5. Morse theory.
   
   
6. Differential forms. de Rham's theorem.
   
   
7. Smales immersion theorem.

Prerequisites

Analysis II (18.101) and Algebraic Topology (18.905)

Grading

100% of the grading is based on the assignments.

Textbooks

Bott, Raoul, R. Bott, and Loring W. Tu. Differential Forms in Algebraic Topology (Graduate Texts in Mathematics; 82). Reprint edition. New York: Springer-Verlag, June 1, 1995. ISBN: 0387906134.

Abraham, Ralph, Jerrold E. Marsden, and Tudor Ratiu. "Manifolds, Tensor Analysis, and Applications." Applied Mathematical Sciences. Vol. 75. New York: Springer Verlag, May 1, 1998. ISBN: 3540967907.

Hirsch, Morris W. "Differential Topology." Graduate Texts in Mathematics. Vol. 33. Reprint edition. New York: Springer-Verlag, November 1, 1988. ISBN: 0387901485.