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 400level course on Abstract Algebra and one on Real Analysis. In reality, the main prerequisite is a solid knowledge of multivariable calculus and linear algebra.
Text: Introduction to Smooth Manifolds by John M. Lee. Course outline
Other helpful reference books:
 Differential Topology by V. Guillemin and A. Pollack  an easyto read book; Chapters 15 give a nice introduction to manifolds.
 Foundations of Differentiable Manifolds and Lie Groups by Frank Warner.
 Lectures on the Geometry of Manifolds by Liviu Nicolaescu.
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 A517. 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 

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 