Math 868 Fall 07

Geometry and Topology 1

Instructors:

Prof. T. Parker
Mathematics Department
A346 Wells Hall
353-8493
parker@math.msu.edu MWF 1:30 -- 2:30
and by appointment
Kwangho Choi
A-527 Wells Hall
884-1469
choikwa4@msu.edu
www.msu.edu/~choikwa4/
Office hours: 3-4 Weds & Thursday.
Math Tools

Goals: This course introduces the intuition and techniques used to study manifolds. Manifolds are the natural setting for calculus in its most appealing and flexible form. They are the primary objects in much of modern geometry and topology. Topics will include differentiable manifolds and tangent spaces, vector bundles, transversality, calculus on manifolds, differential forms, tensor bundles, the Frobenius Theorem, the deRham Theorem and cohomology groups. If time permits, we will cover the Hodge Theorem and the beginnings of Riemannian geometry.

Background: The official prerequisites are a 400-level course on Abstract Algebra and one on Real Analysis. In reality, the main prerequisite is a solid knowledge of multi-variable calculus and linear algebra.

Text: Introduction to Smooth Manifolds by John M. Lee.      Course outline

Other helpful reference books:

Homework:  HW1    HW2    HW3    HW4    HW5    HW6    HW7    Midterm Review    HW8    HW9    HW10   HW11    Hint for HW11    HW12

Math 868 Student Seminar meets Wednesdays 5:15 - 6:15 in Room A-517.    Suggested Seminar Topics

In these seminars, students give informal talks to each other on small topics related to the class material. There should be enough time for everyone in the class to give a talk. If you would prefer not to, you many arrange an alternative project. This will count 10% of the course grade. Here is the schedule for these talks:

Sept 12 Cheryl Balm Real and complex projective spaces
Sept 19 Dan Smith The Grassmann manifold
Sept 26 Alex Theakston Manifolds with boundary
Oct 3 Scott Pelak The Whitney Embedding Theorem
Oct 10 Greg Sulisz Lie Groups
Oct 17 Joe Timmer Lie Algebras
Oct 24 Chris Zin Proof of the Inverse Function Theorem
Oct 31
Kyungbae Park
Transversality
Nov 7 Stefan Progovac Symplectic Manifolds
Nov 14 Ed Morrison Hamiltonian Flows
Nov 21 Sam Otten Morse Theory
Nov 21 Weiwen Gu Lens Spaces
Nov 28 Wei Fan Homogeneous Spaces
Dec 5 Antonio Veloz The Cauchy Integral Formula via forms