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:
- Differential Topology by V. Guillemin and A. Pollack --- an easy-to read book; Chapters 1-5 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 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 |
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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 |