Math 550: Differentiable Manifolds II

David Dumas

University of Illinois at Chicago
Fall 2014

General information

Instructor	David Dumas (ddumas@math.uic.edu)
Lectures	MWF 10am in Taft Hall 309
CRN	37063
Office hours	Monday and Friday 2-3pm
Texts	M. Lee, Introduction to Smooth Manifolds. (Springer GTM, 2012)     → Available online through the UIC library
	M. Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1. (Publish or Perish, 1999)
	F. Warner, Foundations of Differentiable Manifolds and Lie Groups. (Springer GTM, 1983)
	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)

Course description

Building on the foundational material from Math 549, we will discuss several aspects of the geometry and topology of smooth manifolds. Topics will include de Rham theory, distributions and foliations, Lie groups, Lie algebras, vector bundles, principal bundles, and symplectic geometry.

A more detailed topic outline is provided in the course syllabus. The topics may be adjusted to account for time constraints and for the background and preferences of the students.

Course materials

• Course syllabus
• Problem list   -   Last updated

Links and resources

Here we collect citations for further reading about course material (beyond the course texts) and links to relevant online materials.

• Poincaré Lemma for star-shaped domains
  • Olver, Applications of Lie Groups to Differential Equations pp63-67 covers the dilation/Lie derivative approach discussed in Lectures 1 and 2
  • This worksheet from Qiang Guang's algebraic topology course describes same approach through direct calculations (instead of the abstract notion of Lie derivatives and Cartan's formula)
• The Snake Lemma and the cohomology long exact sequence
  • Video clip of the proof from "It's My Turn" (1980 motion picture, IMDB entry).
    (This is a direct link to a WEBM video file; if it doesn't work for you, try the flash-based player or download links here.)
• The Baker-Campbell-Hausdorff formula
  • Bourbaki, Elements of Mathematics: Lie Groups and Lie Algebras, Chapter 2, Section 6 discusses the BCH series.
  • Section 1 of these Lecture notes on Lie algebras by Shlomo Sternberg
  • Blog entry / lecture notes by Terry Tao
• Cohomology of compact Lie groups and Lie algebras
  • The original paper of Chevalley and Eilenberg (AMS Transactions, 1948)
  • Prasolov, Elements of Homology Theory, Chapter 5, Section 6 (specifically pp350-351) describes the isomorphism of H^k(G,R) with the ad-invariant subalgebra of ∧^kLie(G).
  • Knapp, Lie groups, Lie algebras, and cohomology introduces cohomology of Lie algebras in Chapter 4, with a brief dsicussion of the motivation from de Rham theory.
  • Online lecture notes:
    • Cohomology and K-Theory of Compact Lie Groups by Chi-Kwong Fok
    • Cohomology of Lie groups and Lie algebras by Somnath Basu
    • Lie Algebra Cohomology by Brian Weber

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