David Dumas

University of Illinois at Chicago

Spring 2019

Primary text | J.M. Lee, Introduction to Smooth Manifolds. (Springer GTM, 2012)Free ebook through UIC |
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Lectures | MWF at 9am in 317 Taft Hall |

Instructor email | david@dumas.io |

Instructor office hours | MWF 11am |

CRN | 37861 |

- Smooth actions and quotients
- Bundles, connections, and curvature
- Symplectic manifolds

- F. Warner,
*Foundations of Differentiable Manifolds and Lie Groups*. (Springer GTM, 1983) - M. Spivak,
*A Comprehensive Introduction to Differential Geometry*, Volume 1. (Publish or Perish, 1999) (Full text on archive.org) - R. W. Sharpe,
*Differential Geometry: Cartan's Generalization of Klein's Erlangen Program*. (Springer GTM, 2000) - A. Cannas da Silva,
*Lectures on Symplectic Geometry*. (Online lecture notes; also available in Springer LNM series) - D. McDuff and D. Salamon,
*Introduction to Symplectic Topology*. (Oxford, 1999)

**Assignment due on Friday January 18: Read the syllabus****Problem set 1, due Friday February 1**(due date changed due to lecture cancelation)**Problem set 2 due Monday February 11****Problem set 3 due Monday February 25**

- Lecture 1 (Mon Jan 14): Intro: Smooth distributions and the Frobenius theorem (Lee, Chapter 19)
- Lecture 2 (Wed Jan 16): The local Frobenius theorem
- Lecture 3 (Fri Jan 18): Proof of the local Frobenius theorem
- Lecture 4 (Wed Jan 23): The global Frobenius theorem
- Lecture 5 (Fri Jan 25): Proof of the global Frobenius theorem
- Lecture 6 (Mon Jan 28): The Lie correspondence
- Lecture 7 (Wed Jan 30): (class canceled due to weather)
- Lecture 8 (Fri Feb 1): Topological group actions, properness (Lee, Chapter 21)
- Lecture 9 (Mon Feb 4): Smooth actions, the Quotient Manifold Theorem
- Lecture 10 (Wed Feb 6): Slices and tubes
- Lecture 11 (Fri Feb 8): Completion of proof of the QMT
- Lecture 12 (Mon Feb 11): Homogeneous spaces, properly discontinuous group actions
- Lecture 13 (Wed Feb 13): Fiber bundles
- Lecture 14 (Fri Feb 15): G-structures and principal bundles
- Lecture 15 (Mon Feb 18): Principal bundles as right G-spaces, associated bundles, cocycles