Math 285y: Tropical Geometry, Spring 2013
Tuesdays and Thursdays 11:30-1:00, 222 Science Center (note room change)

Instructor: Melody Chan, office 242a Science Center
email: mtchan at
office hours: Thursdays 2:30-3:30 and by appointment
course website:

Course Assistant: Nathan Pflueger, office 242g Science Center
Problem session: Wednesdays 4-5, common room, fourth floor
email: pflueger at

Maclagan and Sturmfels Introduction to Tropical Geometry (textbook in progress).
The first three chapters are available at the link above.
The authors welcome any comments or corrections on the manuscript from you, even small typos. Please e-mail them to Diane Maclagan.

Some survey articles that may be helpful:
Gubler A guide to tropicalizations
Maclagan Introduction to tropical algebraic geometry
Mikhalkin Tropical geometry and its applications
Speyer and Sturmfels Tropical mathematics

This is a one semester graduate topics course on tropical geometry, an emerging field bridging combinatorics, algebraic geometry, and nonarchimedean analytic geometry, with applications to many other areas.

In the first part of the course, we will study "embedded" tropical varieties: given a subvariety of the algebraic torus (K^*)^n, we will define a polyhedral complex in R^n called its tropical variety. Topics include: structure of tropical varieties, the fundamental theorem of tropical geometry, tropical linear algebra and matroid theory, and applications to computations like elimination and implicitization. We will loosely follow the textbook draft by Maclagan and Sturmfels.

The second part of the course will be a more detailed study of "abstract" tropical curves, which are vertex-weighted metric graphs, and their relation to algebraic and nonarchimedean analytic curves; and moduli spaces of tropical curves. We'll discuss many cutting-edge papers in the field; I'll indicate which ones as we go along.

Some experience with algebraic geometry would be extremely helpful for Part 2 of the course. For Part 1, undergraduate algebraic geometry may be enough. An enjoyment of combinatorics, especially polyhedra and graph theory, will be helpful throughout.

If you would like to take this class and don't know whether your background is sufficient, I'd be happy to discuss it with you and try to work something out.


Homework 1, due Thursday February 14
Homework 2, due Tuesday March 5
Homework 3, due Thursday April 11

I encourage you to take this class for a grade if you can. Expect a handful of problem sets during the first part of the course, plus a final project. This project will take the form of a talk on a topic or paper of your choice, to be presented in the final weeks of class, plus a write-up of your lecture.

Even if you are just sitting in on this course, and whether or not you are Harvard-affiliated, I still strongly encourage you to give a talk in class this semester. Please e-mail me to discuss possible topics.

Please take note: no late homework will be accepted, unless you explicitly ask for (and receive) an extension from me.

I urge you to collaborate on all problem sets. Please write up your solutions separately and indicate with whom you collaborated.

Accommodations for students with disabilities
If you need accommodations for a disability, please talk to me as soon as possible and within the first two weeks of the term.

I will try to post the topics to be discussed shortly before each lecture, so that you can plan accordingly.

January 29: Overview: what is tropical geometry?
We did a whirlwind tour of tropical geometry, including: the definition of tropical variety, Newton polygons and Eisenstein's criterion, Mikhalkin's correspondence theorem, divisors on graphs, Riemann-Roch, and specialization.

February 1: Basic definitions and notation for tropicalizations, tropical polynomials, and polytopes. Tropical hypersurfaces and their dual complexes. Generalized Grobner bases and Kapranov's theorem for hypersurfaces over algebraically closed, nontrivially valued fields. References: Textbook sections 2.3, 2.4

February 5: Recap of tropical plane curves, definition of multiplicity of maximal faces. Grobner bases with weights and the proof of Kapranov's theorem for hypersurfaces. References: Textbook sections 2.4, 3.1

February 7: Grobner complexes and the fundamental theorem of tropical geometry. How to compute Grobner complexes. References: Textbook sections 2.4, 2.5

February 12: Definition of tropical basis. Crash course on matroids. Tropicalized linear spaces are Bergman fans of their matroids. References: textbook 2.6 and "The Bergman complex of a matroid and phylogenetic trees" (Ardila and Klivans)

February 14: Every ideal has a tropical basis. Tropicalized linear spaces over valued fields, the tropical Grassmannian, phylogenetic trees, and tropical M_{0,n}. References: Textbook 2.6; David Speyer thesis; "The tropical Grassmannian" (Speyer and Sturmfels)

February 19: Guest lecture, Bernd Sturmfels: The tropical Burkhardt quartic
The Burkhardt quartic is a concrete realization of the moduli space of abelian surfaces with level 3 structure. We compute the tropicalization of this threefold from an embedding into 39-dimensional space. A key player is the finite simple group of order 25920. Tricanonical curves of genus 2 will serve to illustrate the beauty of explicit polynomial equations.

February 21: Nathan Pflueger, Tropical implicitization, tropical compactifications.

February 26: Noether normalization for tori; tropicalization is dimension-preserving. Definition of multiplicities and balancing; statement of the Structure Theorem for tropical varieties. Reference: textbook 3.3, 3.4.

February 28: Multiplicities, balancing, and connectivity in codimension 1. How gfan computes tropical varieties. Definition of complex amoeba and start of discussion of Correspondence Theorem. References: textbook 3.4, 3.5, "Computing tropical varieties" (Bogart-Jensen-Speyer-Sturmfels-Thomas)

March 5: Start the tropical Caporaso-Harris formula. References: "The tropical Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry" (Gathmann and Markwig), "Enumerative tropical algebraic geometry in R^2" (Mikhalkin)

March 7: Part 2 of the tropical Caporaso-Harris formula. References: "The tropical Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry" (Gathmann and Markwig), "Enumerative tropical algebraic geometry in R^2" (Mikhalkin)

March 12: Mikhalkin's Correspondence theorem, part 1. Reference: "Enumerative tropical algebraic geometry in R^2" (Mikhalkin)

March 14: Mikhalkin's Correspondence theorem, part 2, and guest lecture by Vivek Shende.

March 19: Spring break

March 21: Spring break

March 26: Guest lecture, Noah Giansiracusa: Tropical scheme theory

March 28: How to analytify a scheme. Example: the Berkovich affine line. The Berkovich analytification is the inverse limit of tropicalizations. Reference: "Limits of tropicalizations" (Foster, Gross, Payne)

April 2: Semistable reduction for Berkovich curves. Abstract tropical curves, their Jacobians, and the tropical Abel-Jacobi map. Reference: "Tropical curves, their Jacobians, and theta functions" (Mikhalkin and Zharkov)

April 4: Tropical M_{g,n}, A_g, the Torelli theorem, and the Teichmuller perspective. Reference: "On the tropical Torelli map" (Brannetti-Melo-Viviani), "Tropical Teichmuller and Siegel spaces" (Chan-Melo-Viviani)

April 9: Guest lecture, Diane Maclagan

April 11: Part 1 of Metrized complexes of curves: Riemann Roch, specialization, limit linear series and tropical Brill-Noether theory. Reference: "Linear series on metrized complexes of algebraic curves" (Amini and Baker)

April 16: Part 2 of Metrized complexes of curves: Riemann Roch, specialization, limit linear series and tropical Brill-Noether theory. Reference: "Linear series on metrized complexes of algebraic curves" (Amini and Baker), "A tropical proof of the Brill-Noether theorem" (Cools, Draisma, Payne, Robeva)

April 18: Ruthi Hortsch, "A refined Chabauty Coleman Bound and tropical Clifford Theorem" (Katz and Zureick-Brown)

April 23: Netanel Blaer, "What are tropical curves doing in the Gross-Siebert program?"
Guest lecture, Joe Rabinoff on "Nonarchimedean geometry, tropicalization, and metrics on curves" (Baker-Payne-Rabinoff)

April 25: Jennifer Park, Tropical discriminants (Dickenstein-Feichtner-Sturmfels)

April 30: Guest lecture, Joe Rabinoff, "Nonarchimedean geometry, tropicalization, and metrics on curves" (Baker-Payne-Rabinoff) and "Lifting harmonic morphisms of tropical curves, metrized complexes, and Berkovich skeleta" (Amini-Baker-Brugalle-Rabinoff)