Schedule for: 16w5153 - Algebraic, Tropical, and Nonarchimedean Analytic Geometry of Moduli Spaces

Arriving in Oaxaca, Mexico on Sunday, May 1 and departing Friday May 6, 2016
Sunday, May 1
14:00 - 23:59 Check-in begins - open 24 hours (Front desk at your assigned hotel)
19:30 - 22:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
20:30 - 21:30 Informal gathering
A welcome drink will be served at the hotel.
(Hotel Hacienda Los Laureles)
Monday, May 2
07:30 - 09:15 Breakfast (Restaurant at your assigned hotel)
09:15 - 09:30 Introduction and Welcome (Conference Room San Felipe)
09:30 - 10:30 Hannah Markwig: Counting curves in surfaces: the tropical and the Fock space approach
Tropical geometry can be viewed as an efficient tool to organize degenerations. The techniques to count curves in surfaces via tropical geometry are related to the Fock space approach initiated by Cooper-Pandharipande, via floor diagrams (which can be viewed as the combinatorial essence of a tropical curve count) (following Block-Goettsche). Our own contribution relates the tropical and the Fock space approach for descendant Gromov-Witten invariants. (Joint work with Renzo Cavalieri, Paul Johnson and Dhruv Ranganathan.)
(Conference Room San Felipe)
10:30 - 11:30 Coffee Break (Conference Room San Felipe)
11:30 - 12:30 Dhruv Ranganathan: Skeletons of Spaces of Maps and Superabundant Geometries
I will discuss a unified approach to tropicalizing moduli spaces of logarithmic stable maps to arbitrary logarithmically smooth targets using the geometry of Artin-Olsson fans. I will pay particular attention to superabundant case, exploring the relationship between obstructedness of logarithmic maps and the tropical realizability question. Some relatively transparent consequences of the framework include a new perspective on Speyer's well-spacedness condition, polyhedrality of the tropical realizability cycle, and a version of Murphy's Law for lifting problems for lifting tropical curves.
(Conference Room San Felipe)
12:30 - 12:45 Group Photo (Hotel Hacienda Los Laureles)
13:00 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 - 16:00 Martin Ulirsch: Logarithmic structures, Artin fans, and the moduli stack of tropical curves
Artin fans are logarithmic algebraic stacks that are logarithmically \'etale over the base field. Despite their seemingly abstract definition, the geometry of Artin fans can be described completely in terms of combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato’s notion of a fan. In this talk, following a rapid introduction to logarithmic geometry from a modular point of view, I am going to give an expository account of the theory of Artin fans and explain how Thuillier's non-Archimedean skeleton of a toroidal embedding can be understood as an analytification of the associated Artin fan. In the special case of a toric variety, this simply reduces to the fact that the Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient. Finally, Artin fans also provide the motivation for an ongoing joint project with R. Cavalieri, M. Chan, and J. Wise, in which we develop a stack-theoretic framework for the study of tropical moduli spaces.
(Conference Room San Felipe)
16:00 - 16:30 Coffee Break (Conference Room San Felipe)
16:30 - 18:30 Mini-Talks (Conference Room San Felipe)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Tuesday, May 3
07:30 - 09:15 Breakfast (Restaurant at your assigned hotel)
09:30 - 10:30 Ravi Vakil: The geometry of moduli spaces by cutting and pasting
I will discuss past and ongoing work with Melanie Wood on the (stabilization of) topology of various moduli spaces (and “discriminant loci” therein) by writing their classes in the Grothendieck ring in terms of motivic zeta functions (zeta functions in the Grothendieck ring).
(Conference Room San Felipe)
10:30 - 11:30 Coffee Break (Conference Room San Felipe)
11:30 - 12:30 Kristin Shaw: Tropicalizations of Del Pezzo surfaces
In this talk we discuss the structure of tropicalisations of very affine surfaces obtained from del Pezzo surfaces of degree 5, 4, 3 by removing their \((-1)\)-curves. Classically these surfaces are the blow ups of \(\mathbf{P}^2\) in 4, 5, and 6 points (respectively) in general position. The \((-1)\)-curves on the surface then translate to an arrangement of metric trees. The main example will be del Pezzos of degree 3 which are cubic surfaces in \(\mathbf{P}^3\) and the \((-1)\) curves are exactly the 27 lines living on the surface. In this case there are only two non-degenerate combinatorial types of tropicalisations. These combinatorial models are found using three methods; tropicalisation of ideals, fibres of moduli maps and from point configurations in the tropical plane. This talk is based on joint work with Q. Ren and B. Sturmfels
(Conference Room San Felipe)
13:00 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 - 16:00 Andreas Gross: Tropicalizing cycle classes and a correspondence theorem for descendant invariants
Intersection theory is an extremely useful tool both in algebraic and tropical enumerative geometry. It can also be used to obtain algebraic/tropical correspondence theorems. The idea is to apply the tropicalization procedure to appropriate intersection products on suitable algebraic moduli spaces, thus obtaining the analogous tropical intersection products on the analogous tropical moduli spaces. For this to work in general, we need to be able to tropicalize cycle classes. I will present a suitable definition of tropical rational equivalence on extended cone complexes making this possible. As an application we obtain a correspondence theorem for genus 0 logarithmic descendant Gromov-Witten invariants of toric varieties.
(Conference Room San Felipe)
16:00 - 16:30 Coffee Break (Conference Room San Felipe)
16:30 - 17:30 Nathan Pflueger: Brill-Noether-special chains of loops
I will classify the special divisors on a metric graph formed as a chain of loops with arbitrary edge lengths. In particular, I will state a necessary and sufficient criterion for a chain of loops to be Brill-Noether general (in the sense of Cools, Draisma, Payne, and Robeva). As an application, I will state an analog of the Brill-Noether theorem for algebraic curves of fixed gonality, by specializing to chains of loops with special edge lengths.
(Conference Room San Felipe)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Wednesday, May 4
07:30 - 09:00 Breakfast (Restaurant at your assigned hotel)
09:00 - 12:45 Free Morning / Excursion (Oaxaca)
13:00 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 - 16:00 Valery Alexeev: Generalizations of Losev-Manin spaces to other root systems
The Losev-Manin space parameterizing stable rational curves with the n+2 points weighted \(1^2, \epsilon^n\)​ is naturally associated with the root system of type \(A_{n-1}\)​. I will explain generalizations of this moduli space to the other root systems of ADE and extended ADE types. The talk is based on a joint work with Alan Thompson.
(Conference Room San Felipe)
16:00 - 16:30 Coffee Break (Conference Room San Felipe)
16:30 - 17:30 Jeffrey Giansiracusa: Matroids and exterior algebras in tropical algebra
Classically, the exterior algebra provides an elegant description of the Plucker embedding.  In this talk I will present an analogue of this picture for matroids (and valuated matroids). Matroids can be thought of as objects defined over the boolean idempotent semiring B={0,1}.  We define a tropical analogue of exterior algebra for any quotient the free module B^n, and we show that a d-multivector w is a matroid if and only if the quotient of B^n that is dual to the kernel of wedging with w has d-th wedge power free of rank 1.
(Conference Room San Felipe)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Thursday, May 5
07:30 - 09:15 Breakfast (Restaurant at your assigned hotel)
09:30 - 10:30 Soren Galatius: Moduli spaces of graphs
A metric graph is a graph together with an assignment of a positive real number to each edge. The isometry classes of metric graphs are points in a moduli space of metric graphs. Differences in the detailed definition leads to several flavors of moduli spaces of graphs with quite different homotopy types. Some flavors are related to automorphism groups of free groups, and some are related to moduli space of curves and tropical geometry. I will survey the different moduli spaces and the various maps between them, from a homotopy theoretic viewpoint. If time permits, I will discuss some new results joint with Melody Chan and Sam Payne.
(Conference Room San Felipe)
10:30 - 11:30 Coffee Break (Conference Room San Felipe)
11:30 - 12:30 Yuji Odaka: Tropical geometric compactification of moduli varieties
We compactify moduli varieties of (un-embedded, polarized) varieties whose boundaries are the corresponding tropical moduli space (hence “cell complex”-like) i.e. the moduli of the corresponding class of tropical varieties. Cf., arXiv:1406.7772 
(Conference Room San Felipe)
13:00 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)
15:00 - 16:00 Tyler Foster: Analytic and tropical varieties over higher rank valued fields
Recent work of Nisse-Sottile, Hrushovski-Loeser, Ducros, and Giansiracusa-Giansiracusa demonstrates that valuation rings of rank >1 play an important role in the geometry of analytic and tropical varieties over non-Archimedean valued fields of rank 1. In this talk, I will present recent work with Dhruv Ranganathan and Max Hully on the geometry of analytic and tropical varieties over higher rank valued fields. I will explain how ideas coming from tame topology can be used to study the geometry of these spaces. I will explain the role that multi-stage degenerations play in this geometry, and if time permits, I will sketch an application of these ideas to the study of tropical and algebraic intersection multiplicities.
(Conference Room San Felipe)
16:00 - 16:30 Coffee Break (Conference Room San Felipe)
16:30 - 17:30 Johannes Nicaise: Refined tropical curve counting and motivic integration
Motivated by mathematical physics, Göttsche and Shende have defined "quantized" versions of certain enumerative invariants for linear systems on surfaces. Block and Göttsche have proposed formulas for the corresponding multiplicities for tropical curves, refining the classical tropical multiplicities. In certain cases, they were able to prove a correspondence theorem for these refined invariants. In joint work with Sam Payne and Franziska Schroeter, we suggest a geometric interpretation of the Block-Göttsche multiplicities as \(\chi_y\) genera of semi-algebraic analytic domains over the field of Puiseux series. In order to define and compute these \(\chi_y\) genera, we use the theory of motivic integration developed by Hrushovski and Kazhdan and we explore its connections with tropical geometry.
(Conference Room San Felipe)
19:00 - 21:00 Dinner (Restaurant Hotel Hacienda Los Laureles)
Friday, May 6
07:30 - 09:15 Breakfast (Restaurant at your assigned hotel)
10:30 - 11:30 Coffee Break (Conference Room San Felipe)
13:00 - 15:00 Lunch (Restaurant Hotel Hacienda Los Laureles)