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 |
---|---|

Lectures | MWF at 9am in 317 Taft Hall |

Instructor email | david@dumas.io |

Instructor office hours |
MWF 11am (May 3 office hours end at 11:25am) |

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) - C. Taubes,
*Differential Geometry: Bundles, Connections, Metrics and Curvature*. (Oxford, 2011) - S. Kobayashi, K. Nomizu,
*Foundations of Differential Geometry, Volume 1*(Wiley, 1996)

(This is a reprint of the 1963 edition, which can also be used.)

**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****Problem set 4 due Monday March 11****Problem set 5 due Wednesday April 3**Updated March 17**Problem set 6 due Monday April 15****Problem set 7 due Monday April 29**

- 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
- Lecture 16 (Wed Feb 20): The frame bundle and associated bundles
- Lecture 17 (Fri Feb 22): Reduction of structure group
- Lecture 18 (Mon Feb 25): Smooth path connections, groupoids
- Lecture 19 (Wed Feb 27): Principal connections as splittings of
*TP*and as 𝔤-valued forms - Lecture 20 (Fri Mar 1): Horizontal lifts
- Lecture 21 (Mon Mar 4): Local connection forms
- Lecture 22 (Wed Mar 6): More on Maurer-Cartan forms; existence of principal connections
- Lecture 23 (Fri Mar 8): Curvature of principal connections
- Lecture 24 (Mon Mar 11): Parallel transport in principal bundles
- Lecture 25 (Wed Mar 13): Connections on vector bundles
- Lecture 26 (Fri Mar 15): Parallel transport in vector bundles
- Lecture 27 (Mon Mar 18): Curvature in vector bundles
- Lecture 28 (Wed Mar 20): The exterior covariant derivative
- Lecture 29 (Fri Mar 22): Curvature is the derivative of parallel transport
- Lecture 30 (Mon Apr 1): Equivalent characterizations of flatness
- Lecture 31 (Wed Apr 3): Correspondence between flat vector bundles and twisted products
- Lecture 32 (Fri Apr 5): Equivalence of vector and principal connections
- Lecture 33 (Mon Apr 8): Equivalence of vector and principal connections (continued)
- Lecture 34 (Wed Apr 10): Symplectic vector spaces
- Lecture 35 (Fri Apr 12): Symplectic manifolds
- Lecture 36 (Mon Apr 15): Hamiltonian vector fields and the Poisson bracket
- Lecture 37 (Wed Apr 17): The Darboux Theorem
- Lecture 38 (Fri Apr 19): Examples of symplectic manifolds and Hamiltonian systems
- Lecture 39 (Mon Apr 22): Symplectic and Hamiltonian group actions
- Lecture 40 (Wed Apr 24): Examples of moment maps
- Lecture 41 (Fri Apr 26): Marsden-Weinstein-Meyer quotient theorem
- Lecture 42 (Mon Apr 29): Student presentations
- Lecture 43 (Wed May 1): Student presentations
- Lecture 44 (Fri May 3): Student presentations

- Groupoids
- Briefly mentioned in S. MacLane,
*Categories for the Working Mathematician*, which is a good general resource if categorical algebra arises in your work but is not your main focus area. - This nice survey discusses applications and has many references: R. Brown,
*From Groups to Groupoids: A Brief Survey*, Bulletin of the London Mathematical Society 19 (1987) 113-134.

- Briefly mentioned in S. MacLane,
- Path groupoid, path connections, thin homotopy
- My treatment in lecture most closely follows: E. E. Wood,
*Reconstruction theorem for groupoids and principal fiber bundles*, International Journal of Theoretical Physics 36 (1997) 1253-1267.^{*} - Smooth thin homotopy was introduced and studied in relation to connections in: A. Caetano and R. F. Picken,
*An Axiomatic Definition of Holonomy*, International Journal of Mathematics 5 (1994) 835-848.. (A preprint version is available in full text from CERN.)

- My treatment in lecture most closely follows: E. E. Wood,
- Equivalence of vector and principal connections
- Sections 18 and 19 of: P. W. Michor,
*Topics in Differential Geometry*, American Mathematical Society, 2008.

- Sections 18 and 19 of: P. W. Michor,