Graduate Student Seminar, Fall 2014
Professor A.J. de Jong,
Columbia university,
Department of Mathematics.
Organizational:
 Please email me if you want to be on the associated mailing list.
 The talks will be roughly 45+15+30 minutes where 15 = break.
 Time and place: Fridays 10:30  12:00 AM in Room 307.
 First organizational meeting: Friday, Sept 5.
 No lecture on: Firday, Nov 28.
 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].
 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.
 Sept 19. Remy van Dobben de Bruyn. Ranks of divisors and the tropical RiemannRoch theorem. Section 2.2. Prove the RiemanRoch theorem in this setting. Fun! Link to online notes.
 Sept 26. Natasha Potashnik. Reduced divisors and Luo's Theorem. Section 2.3. Prove Luo's theorem.
 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.
 Oct 10. Raju Krishnamoorthy. Deduce Corollary 1.2 from Theorem 1.1. Section 3.
 Oct 17. Daniele Turchetti. First part of proof of Theorem 1.1.
 Oct 24. Johan de Jong. Second part of proof of Theorem 1.1.
 Oct 31. AGNES; no talk this week.
 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.
 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 SpeyerSturmfels.
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.
 Nov 21. Dmitry Zakharov. Tropical integral systems
 Nov 28. Thanksgiving; no talk this week
 Dec 5. Johan de Jong. Talk about the paper The tropicalization of the moduli space of curves
by Abramovich, Caporaso, Payne. link
 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
 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.
 Further topics: Relationship Berkovich spaces and tropical geometry.
For example the papers 1404.0279
and the earlier 1104.0320.
 Further topics: On a CohenLenstra heuristic for Jacobians of
random graphs, J. Clancy, N. Kaplan, T. Leake, S. Payne, and M. Wood.
link
References
In random order!
 [CDPR] Cools, Filip; Draisma, Jan; Payne, Sam; Robeva, Elina,
A tropical proof of the BrillNoether theorem,
Adv. Math. 230 (2012), no. 2, 759  776.
link
 [B] Baker, Matthew,
Specialization of linear systems from curves to graphs
Algebra Number Theory 2 (2008), no. 6, 613  653.
link
 [BN] Baker, Matthew; Norine, Serguei,
RiemannRoch and AbelJacobi theory on a finite graph,
Adv. Math. 215 (2007), no. 2, 766  788.
link
 [C] Caporaso, Lucia,
Algebraic and combinatorial BrillNoether theory,
Contemp. Math., 564, Amer. Math. Soc., Providence, RI, 2012.
link
 [GK] Gathmann, Andreas; Kerber, Michael,
A RiemannRoch theorem in tropical geometry,
Math. Z. 259 (2008), no. 1, 217  230.
link
 [Mz] Mikhalkin, Grigory; Zharkov, Ilia,
Tropical curves, their Jacobians and theta functions,
Contemp. Math., 465, Amer. Math. Soc., Providence, RI, 2008.
link
 [BF] Baker, Matthew; Faber, Xander,
Metric properties of the tropical AbelJacobi map,
J. Algebraic Combin. 33 (2011), no. 3, 349  381.
link
 [JP] Jensen, David; Payne, Sam,
Tropical multiplication maps and the GiesekerPetri theorem,
see this
link
 Webpage
of Matt Baker has lots of relevant papers

Webpage of Lucia Caporaso has lots of relevant papers
 Webpage of
Sam Payne has lots of relevant papers