MAT4520 - Spring 2022

Llog (= lecture log)

  • 17/1: Pages 1-3 of introductory notes.
  • 19/1: Pages 4-5 of introductory notes; section 1 and subsection 2.1 from textbook.
  • 24/1: Rest of section 2; subsections 3.1, 3.3 from textbook.
  • 26/1: Subsections 3.2, 3.4-3.9 from textbook.
  • 31/1: Rest of section 3; subsections 4.1-4.3.
  • 2/2: Rest of section 4; subsections 5.1-5.2.  I often write (U, x) in place of (U, \phi) for a coordinate chart.
  • 7/2: Rest of section 5; subsections 6.1-6.2.
  • 9/2: Rest of section 6.  I avoid the abbreviation F^i for y^i F on page 67, since it does not exhibit the dependence on y.  In the definition of local diffeomorphism on page 68, F(U) should be assumed to be open in M.
  • 14/2: Section 7.
  • 16/2: Sections 8.1-8.6.
  • 21/2: Rest of section 8.  Sections 9.1-9.2.
  • 23/2: Rest of section 9.  Discussion of measure zero and Sard's theorem.  Sections 10.1-10.2.
  • 28/2: Rest of section 10. Sections 11.1-11.2.
  • 2/3: Problem 9.7.  Section 11.3.
  • 7/3: Rest of section 11.  Sections 12.1-12.2.
  • 9/3: Discussion of vector bundles.  Sections 12.3-12.5.  See notes on vector bundles.
  • 14/3: Section 13.
  • 16/3: Sections 14.1-14.4.  Existence of solutions to an ODE by a contraction operator.
  • 21/3: Section 14.5-14.6 and 15.1-15.4.
  • 23/3: Sections 15.5 and 16.
  • 28/3: Sections 17.1-17.5.
  • 30/3: Sections 17.6 and 18.1-18.6.
  • 4/4: Proposition 19.7.  Sections 18.7-18.8 and 19.1-19.4.
  • 6/4: Sections 19.5-19.7 and 20.1-20.2.
  • 20/4: Problems 17.4 and 18.9.  Sections 20.3-20.6 and 21.1.
  • 25/4: Sections 21.2-21.5.
  • 27/4: Problem 19.11. Sections 22.1-22.4.  I incorrectly claimed that C^\infty_p(R^n) restricts isomorphically to C^\infty_p(H^n), but Example 1.3 shows this is false.  The induced map of point derivations T_p H^n --> T_p R^n is nonetheless an isomorphism.
  • 2/5: Sections 22.5-6 and 23.1-4.  See notes on Stokes' theorem.
  • 4/5: Problem 20.9.  Sections 23.5-6 and 24.1-2.
  • 9/5: Sections 24.3-24.4 and 25.
  • 11/5: Sections 26 and 27.1.
  • 16/5: Sections 27.2-27.4.
  • 18/5: Problem 25.3.  Cohomology of the n-sphere.  Sections 28.1-28.2.
  • 23/5: Sections 28.3 and 29 (for M=U open in R^n). Statement of Poincare duality and the Kuenneth theorem.
Published Jan. 26, 2022 3:00 PM - Last modified May 23, 2022 2:08 PM
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