MATH 655
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Differential Topology
Mathematics
College of Computational, Mathematical, & Physical Sciences
Course Description
An introduction to smooth manifolds and their topology: smooth manifolds; tangent, vector, and cotangent bundles; immersions, submersions, and embeddings; tubular neighborhoods; transversality; differential forms, integration, and Stoke's Theorem; deRham cohomology; and degree theory.
When Taught
Fall Even Years
Min
3
Fixed/Max
3
Fixed
3
Fixed
0
Other Prerequisites
Math 342 or equivalent; Math 553 or equivalent.
Recommended
Math 554
Title
Overview
Learning Outcome
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
Title
Learning Outcomes
Learning Outcome
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. For more detailed information visit the Math 655 Wiki page.
