Summer Graduate School

A tropical stable map and the corresponding floor diagram

Enumerative geometry and the theory of moduli spaces of curves are two cornerstones of modern algebraic geometry; the two subjects have had a significant influence on each other. In the last 15 years, discrete and combinatorial methods, systematized within tropical geometry, have begun to provide new avenues of access into these two subjects. The goal of this summer school is to give students crash courses in tropical and logarithmic geometry, with a particular focus on the applications in enumerative geometry and moduli theory.  The school will consist of three courses of seven lectures each:

1. Enumeration of tropical curves/ by Hannah Markwig
2. Curve counting in tropical and algebraic geometry by Renzo Cavalieri
3. Logarithmic geometry and stable map/s by Dhruv Ranganathan

By combining partial results from all three lecture series, students will obtain a deep connection between logarithmic Gromov–Witten invariants and tropical curves. Such a connection has some compelling outcomes in geometry more broadly. For instance, a study of the tropical realizability problem has deep applications to Brill–Noether theory and the classical geometry of curves; tropical refined curve counting through logarithmic Gromov–Witten theory has immediate applications to quantizations of mirror manifolds.  In summary, the school will offer a unique blend of modern techniques in Gromov–Witten theory and combinatorial algebraic geometry.

School Structure
There will be lectures and problem sessions. The purpose of the problem sessions is to reinforce and deepen students’ understanding of the material from the lectures by working on problems and in certain cases, to discuss material relevant to future lectures. This will be done by discussion, question-and-answer, and problem-solving from problem sets. Activities may include presentation of solutions to problems assigned based on lectures, student or TA presentation of material relevant to lecture, or exercises and presentation of relevant background material.

Suggested Prerequisites
The participants will be expected to have some basic understanding of algebraic geometry. Also, an inclination towards combinatorics would be suitable. The following will certainly be sufficient:

• Some knowledge of algebraic geometry at the level of the first chapters of books by Hartshorne or Shafarevich. A first course in scheme theory is certainly more than what is required.
• It will be helpful, though not necessary, to have some familiarity with toric varieties (such as the first 2-3 chapters of Cox, Little, Schenk - Toric varieties): the basic statements about their structure and an understanding of basic examples.
• While also not necessary, some familiarity with Chern classes (for instance in a topological context), will help. A useful reference is "Moduli spaces of pointed rational curves", lecture notes for the Combinatorial Algebraic Geometry Program at the Fields Institute, https://www.math.colostate.edu/~renzo/teaching/Moduli16/Fields.pdf.

Participants who wish to gain a sense for the flavour of and motivation for enumerative geometry might consult the introductory text Kontsevich’s Formula for Rational Plane Curves by Kock and Vainsencher. Participants who want to gain a sense for the flavor of tropical geometry methods may check the paper "A bit of tropical geometry" by Brugalle-Shaw (Amer. Math. Monthly 121 (2014), no. 7, 563–589, https://arxiv.org/abs/1311.2360).

Additional Reading (all in Arxiv)

• Brugallé & Shaw, 2014, A bit of tropical geometry
• Richter-Gebert et al., 2003, First steps in tropical geometry
• Cavalieri et al., 2010, Tropical Hurwitz numbers
• Cavalieri et al., 2016, Tropical compactification and the Gromov--Witten theory of P^1
• Gathmann & Markwig, 2008, Kontsevich's formula and the WDVV equations in tropical geometry