Graduate Student Seminar, Spring 2014

Professor A.J. de Jong, Columbia university, Department of Mathematics.

Organizational:

1. Please email me if you want to be on the associated mailing list.
2. The talks will be 45+15+30 minutes where 15 = break.
3. Time and place: Fridays 10:30 -- 12:00 AM in Room 312.
4. First lecture: Friday January 24, 2014.
5. No lecture on: Mar 21.

Schedule (we may still move around some of the talks)

• Jan 31. Yang An. Casas-Alvero conjecture
• Feb 7. Xuanyu Pan. Stephanov-Bombieri method to prove Weil conjecture
• Feb 14. Ashwin Deopurkar. Theta characteristics
• Feb 21. NO LECTURE TODAY. TALKS MOVED 1 WEEK.
• Feb 28. Philip Engel. Sets of zero coefficients of algebraic power series
• Mar 7. Natasha Potashnik. No Jordan-Holder in D^b(X)
• Mar 14. Dingxin Zhang. Phantom categories
• Mar 21. SPRING RECESS -- NO LECTURE
• Mar 28. Raju Krishnamoorthy. Recognizing Shimura varieties
• April 4. Xia Jie. Tensor products of semi stable
• April 11. Michael Thaddeus. Eigenvalues of sums and products of matrices
• April 18. Johan de Jong. Honeycombs and saturation conjecture
• April 25. AGNES at Stony Brook -- no seminar.
• May 2. Remy van Dobben de Bruyn. Tannakian categories

List of topics: The goal of each lecture is to state (one or more) result from the paper(s) and to highlight (at least) one key new idea or argument.

1. Find your own topic. Choose a fun and interesting paper from the literature and lecture about it in the seminar.
2. Borel-Weil-Bott: A very simple proof of Bott's theorem Une démonstration algébrique d'un théorème de Bott
3. The saturation conjecture: The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture, The saturation conjecture,
4. Tannakian categories and representation categories: Tannakian Categories. P. Deligne and J.S. Milne, Catégories tannakiennes
5. Tensor products of semi-stables are semi-stable: Tensor products in p-adic Hodge theory, Diophantine approximations on projective spaces
6. Sets of zero coefficients of algebraic power series: On vanishing coefficients of algebraic power series over fields of positive characteristic, A Skolem¿Mahler¿Lech theorem in positive characteristic and finite automata, Suites algébriques, automates et substitutions
7. Phantom categories: Geometric Phantom Categories, Determinantal Barlow surfaces and phantom categories,
8. No Jordan-H\"older for derived categories: On the Jordan-Hölder property for geometric derived categories,
9. Kazhdan, Varshavsky and Shimura varieties: On the characterization of complex Shimura varieties (interesting to read Milne's review of this paper on mathscinet), Kazhdan's papers On arithmetic varieties and On arithmetic varieties II.
10. Talk about Stephanov's method to prove the Weil conjecture for curves as in Bombieri's talk in the Bourbaki seminar Counting points on curves over finite fields.
11. TBA

The following are somehow fun and a bit of a different nature.

12. The Casas-Alvero conjecture: The Casas-Alvero conjecture for infinitely many degrees
13. The Jacobian conjecture: Go to the website of Arno van der Essen and look around. Choose one of the most fun results of the type "the Jacobian conjecture is equivalent to or implies by Conjecture ???" and explain it in a lecture.