Math 8306-07

List of Topics of Presentations

As you all know, each of you should make a topic presentation during the school year. 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 not from this list - talk to me about that.

  1. Cohomology of SO(n) (Section 3.D) - Claimed by Eric for October 28
  2. The Gysin sequence and the cohomology of BSO(n) (Section 4.D, or Husemoller or Milnor-Stasheff)
  3. Vector fields and the Euler characteristic (Milnor, Topology from the Differentiable Viewpoint, Section 6; or any other source) - Claimed by Jeongoo Cheh for January 24
  4. Poincare Duality (Section 3.3) Claimed by myself for October 30
  5. The Leray-Hirsch Theorem (Section 4.D) - Claimed by Sangwook Kim for May 2
  6. Sheaf cohomology (Griffiths-Harris, Section 0.3) -Claimed by Fei Xu for November 15
  7. The Eilenberg-Zilber Theorem (R. Switzer, Algebraic Topology..., Chapter 13) - Claimed by myself for the Fall term
  8. The James construction and the homology of the loop space of a suspension (Section 4.J) - Claimed by James for February 17
  9. Simplicial approximation (Section 2.C)
  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.) - Claimed by Kuerak for May 5
  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) - Claimed by Javier for April 2
  16. The Adams spectral sequence (McCleary's Guide, Chapter 9)
  17. The Atiyah-Hirzebruch spectral sequence (McCleary's Guide, Section 11.3)
  18. Chern classes after Grothendieck (Hatcher, Vector Bunldes, Section 3.1)
  19. Curvature and characteristic classes (Appendix to Milnor-Stasheff's book)
  20. The Atiyah-Singer Index Theorem (Section 5.4 of Luke-Mishchenko's book)
  21. The Riemann-Roch Theorem (Section V.4 of M. Karoubi's book)
  22. Postnikov towers (Hatcher or J. P. May, A Concise Course in Algebraic Topology) - Claimed by Jon for May 9
  23. Classifying spaces of posets and categories (Goerss and Jardine, Simplicial homotopy theory) - Claimed by Muge for May 7
  24. De Rham cohomology with compact supports and relative de Rham cohomology (Section V.4 of Karoubi, M.; Leruste, C. Algebraic topology via differential geometry. London Mathematical Society Lecture Note Series, 99. Cambridge University Press, Cambridge, 1987) - Claimed by Hongjie Dong for April 28