Difference between revisions of "Math 655: Differential Topology"
(→Prerequisite) 
m (moved Math 655 to Math 655: Differential Topology) 
(No difference)

Revision as of 16:46, 3 April 2013
Contents
Catalog Information
Title
Differential Topology
Credit Hours
3
Prerequisite
342 or equivalent, and 554 or equivalent.
Description
An introduction to manifolds and smooth manifolds and their topology.
Desired Learning Outcomes
Prerequisites
Knowledge of basic point set topology from Math 553, 554 will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. Basic knowledge of linear algebra (Math 313) and introductory analysis (Math 341 and 342) will also be assumed.
Minimal learning outcomes
Outlined below are topics that all successful Math 655 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.
 Manifolds
 Topological and smooth manifolds
 Manifolds with boundary
 Tangent vectors
 Tangent bundles
 Vector bundles and bundle maps
 Cotangent bundles
 Submanifolds
 Submersions, immersions, embeddings
 Inverse and implicit function theorems
 Transversality
 Embedding and approximation theorems
 Differential forms and tensors
 Wedge product
 Exterior derivative
 Orientations
 Stoke's Theorem
Additional topics
At the discretion of the instructor as time allows. Topics might include Lie groups and homogeneous spaces, Morse functions, de Rham cohomology and the de Rham theorem, Jordan curve theorem, Lefschetz fixedpoint theory, degree, GaussBonnet theorem, etc.