Graduate Student Seminar, Fall 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 roughly 45+15+30 minutes where 15 = break.
  3. Time and place: Fridays 10:30 -- 12:00 AM in Room 307.
  4. First organizational meeting: Friday, Sept 5.
  5. No lecture on: Firday, Nov 28.
  6. Last meeting: Friday, Dec 12.

List of talks

As you can see we still need 2 speakers for talks in the seminar. Please let me know if you are willing to give one of the talks. Section and theorem numbers refer to [CDPR].

  1. Sept 12. Ashwin Deopurkar. Divisors and linear equivalence. Section 2.1. Show that one can reasonably define the Picard group of a metric graph and describe the group of classes of divisors.
  2. Sept 19. Remy van Dobben de Bruyn. Ranks of divisors and the tropical Riemann-Roch theorem. Section 2.2. Prove the Rieman-Roch theorem in this setting. Fun! Link to online notes.
  3. Sept 26. Natasha Potashnik. Reduced divisors and Luo's Theorem. Section 2.3. Prove Luo's theorem.
  4. Oct 3. Qixiao Ma. Baker's Specialization Lemma. Section 2.4. Prove the specialization lemma. This talk will require algebraic geometry for the first time.
  5. Oct 10. Raju Krishnamoorthy. Deduce Corollary 1.2 from Theorem 1.1. Section 3.
  6. Oct 17. Daniele Turchetti. First part of proof of Theorem 1.1.
  7. Oct 24. Johan de Jong. Second part of proof of Theorem 1.1.
  8. Oct 31. AGNES; no talk this week.
  9. Nov 7. Dhruv Ranganathan.
    Title: Berkovich skeletons and the enumerative geometry of target curves
    Abstract: Berkovich analytification captures the (nonarchimedean) analytic geometry of a variety X over a valued field. In various circumstances, the analytification admits a deformation retract to a finite simplicial complex with integral structure, known as the skeleton. When X is a moduli space, the skeleton can often be interpreted as a moduli space for certain objects in tropical geometry. After introducing Berkovich spaces, and their skeletons, I will discuss the geometry of the skeleton of compactifications of spaces of stable maps from curves to curves, and consequences for enumerative geometry. The latter is based on recent and ongoing joint work with Renzo Cavalieri and Hannah Markwig.
  10. Nov 14. Maria Angelica Cueto.
    Title: Faithful tropicalization of the Grassmannian of planes
    Abstract:In this talk, we show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in the sense of Berkovich. Our proof is constructive and it relies on the combinatorial description of the tropical Grassmannian as a space of phylogenetic trees by Speyer-Sturmfels.
    In the first half of my talk, I will discuss the necessary background on analytification of algebraic varieties and the combinatorics of the aforementioned space of trees inside tropical projective space. The second half will be devoted to the proof of the main theorem and its interpretation in terms of the tropicalization map: its fibers are affinoid domains with a unique Shilov boundary point. Our homeomorphism identifies each point in the tropical Grassmannian with the Shilov boundary point on its fiber.
    This is joint work with M. Haebich and A. Werner.
  11. Nov 21. Dmitry Zakharov. Tropical integral systems
  12. Nov 28. Thanksgiving; no talk this week
  13. Dec 5. Johan de Jong. Talk about the paper The tropicalization of the moduli space of curves by Abramovich, Caporaso, Payne. link
  14. Dec 11. Remy van Dobben de Bruyn
    Title: A lower bound for gonality on graphs
    Abstract: Upper bounds for gonality can be found by exhibiting specific divisors; finding good lower bounds is much harder. A recent preprint by J. van Dobben de Bruyn and D. Gijswijt proves that the gonality of a graph is bounded below by the treewidth (a combinatorial invariant).

The following topics were not chosen

  1. Further topics: Tropicalization of algebraic varieties. For example see the general discussion in the paper 1207.1925. Other papers: Geometry in the tropical limit and What is a tropical curve? and Tropical geometry and its applications.
  2. Further topics: Relationship Berkovich spaces and tropical geometry. For example the papers 1404.0279 and the earlier 1104.0320.
  3. Further topics: On a Cohen-Lenstra heuristic for Jacobians of random graphs, J. Clancy, N. Kaplan, T. Leake, S. Payne, and M. Wood. link

References

In random order!

  1. [CDPR] Cools, Filip; Draisma, Jan; Payne, Sam; Robeva, Elina, A tropical proof of the Brill-Noether theorem, Adv. Math. 230 (2012), no. 2, 759 -- 776. link
  2. [B] Baker, Matthew, Specialization of linear systems from curves to graphs Algebra Number Theory 2 (2008), no. 6, 613 -- 653. link
  3. [BN] Baker, Matthew; Norine, Serguei, Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math. 215 (2007), no. 2, 766 -- 788. link
  4. [C] Caporaso, Lucia, Algebraic and combinatorial Brill-Noether theory, Contemp. Math., 564, Amer. Math. Soc., Providence, RI, 2012. link
  5. [GK] Gathmann, Andreas; Kerber, Michael, A Riemann-Roch theorem in tropical geometry, Math. Z. 259 (2008), no. 1, 217 -- 230. link
  6. [Mz] Mikhalkin, Grigory; Zharkov, Ilia, Tropical curves, their Jacobians and theta functions, Contemp. Math., 465, Amer. Math. Soc., Providence, RI, 2008. link
  7. [BF] Baker, Matthew; Faber, Xander, Metric properties of the tropical Abel-Jacobi map, J. Algebraic Combin. 33 (2011), no. 3, 349 -- 381. link
  8. [JP] Jensen, David; Payne, Sam, Tropical multiplication maps and the Gieseker-Petri theorem, see this link
  9. Webpage of Matt Baker has lots of relevant papers
  10. Webpage of Lucia Caporaso has lots of relevant papers
  11. Webpage of Sam Payne has lots of relevant papers