Math 8306-07

List of Topics of Presentations

As you all know, each of you should make a topic presentation during the Spring Term. Below is a list of suggested topics. Sections refer to Hatcher's textbook, unless a different source is mentioned. If you find a source other than the one I suggest, you are welcome to use it. The topics below have different length and level of difficulty. Keep in mind, that you have to make a 50 minute presentation, similar to a talk in a research seminar. You will have to choose only the most important/interesting points of your topic at your discretion and briefly sketch some proofs of main results or ideas beyond them and place all this within 50 minutes.

Choose a topic and tell me, so that I will mark it unavailable and discuss the date of your presentation with you. You may also choose a topic of your interest not from this list - talk to me about that.

  1. Cohomology of SO(n) (Section 3.D)
  2. Basepoints and homology (Section 4.A)
  3. The Gysin sequence and the cohomology of BSO(n) (Section 4.D, or Husemoller or Milnor-Stasheff)
  4. Vector fields and the Euler characteristic (Milnor, Topology from the Differentiable Viewpoint, Section 6; or any other source)
  5. The Leray-Hirsch Theorem (Section 4.D)
  6. Sheaf cohomology (Griffiths-Harris, Section 0.3)
  7. The Eilenberg-Zilber Theorem (R. Switzer, Algebraic Topology..., Chapter 13)
  8. The James construction and the homology of the loop space of a suspension (Section 4.J)
  9. Simplicial approximation (Section 2.C) [Staked out by Joao for May 6]
  10. Bott periodicity (Hatcher, Vector Bundles and K-Theory, Section 2.2, or May, Course in Algebraic Topology, Section 24.2)
  11. Currents and their cohomology (Griffiths-Harris, Section 3.2)
  12. The Lefschetz formula (Griffiths-Harris, Section 3.4)
  13. Cohomology of loop spaces (R. Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology; J.D.S. Jones, Cyclic homology and equivariant homology, Invent. Math., 87 (1987), 403-423.)
  14. Configuration spaces with summable labels and cohomology of free loop spaces (D. Burghelea, The free loop space. I. Algebraic topology. Algebraic topology (Evanston, IL, 1988), 59--85, Contemp. Math., 96, Amer. Math. Soc., Providence, RI, 1989.)
  15. Cohomology of configuration spaces (original papers of Arnold, Mat. Zam., and Fadell-Neuwirth, Math. Scand. 10 (1962) 111-118)
  16. The Adams spectral sequence (McCleary's Guide, Chapter 9) [Staked out by Chris for May 6, 3:35, VinH 213]
  17. The Atiyah-Hirzebruch spectral sequence (McCleary's Guide, Section 11.3)
  18. Chern classes after Grothendieck (Hatcher, Vector Bundles, Section 3.1)
  19. Curvature and characteristic classes (Appendix to Milnor-Stasheff's book) [Staked out by Josef for May 2]
  20. The Atiyah-Singer Index Theorem (Section 5.4 of Luke-Mishchenko's book) [Staked out by Antoine for May 4]
  21. The Riemann-Roch Theorem (Section V.4 of M. Karoubi's book on K-theory) [Staked out by Wenliang for April 29]
  22. Postnikov towers (Hatcher or J. P. May, A Concise Course in Algebraic Topology)
  23. Classifying spaces of posets and categories (Goerss and Jardine, Simplicial homotopy theory) [Staked out by Dan for May 2, 3:35, VinH 311]