Algebraic, Tropical, and Nonarchimedean Analytic Geometry of Moduli Spaces (16w5153)


(Yale University)

Matthew Baker (Georgia Tech)

(Brown University)

(University of Kentucky)


The Casa Matemática Oaxaca (CMO) will host the "Algebraic, Tropical, and Nonarchimedean Analytic Geometry of Moduli Spaces" workshop from May 1st to May 6th, 2016.

Algebraic geometry aims to understand the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations, by studying the interplay between algebraic properties of the system of equations and geometric properties of the solution set. Tropical geometry, on the other hand, is a vast generalization of the classical theory of Newton polyhedra, which allows the study of algebraic varieties over valued fields via polyhedral methods. Nonarchimedean geometry is yet another method for studying algebraic varieties over valued fields, with techniques that are further from polyhedral combinatorics and closer to analysis and number theory. The past half dozen years have seen a series of breakthroughs relating these three fields, with nonarchimedean analytifications of algebraic varieties identified with limits of tropicalizations, specialization lemmas explicitly relating tropical invariants to algebraic invariants, and the introduction of hybrid objects such as metrized complexes of curves that combine tropical geometric objects with algebraic varieties. Most recently, a series of papers have appeared demonstrating that skeletons of nonarchimedean analytic moduli spaces can often be interpreted as moduli spaces of tropical objects, and these tropical modular interpretations have already led to new results in algebraic geometry, such as the computation of new cohomology classes on moduli spaces of curves, as well as more conceptual proofs of older results relating tropical enumerative invariants to classical enumerative invariants in algebraic geometry. This workshop will focus on the interactions between algebraic, tropical, and nonarchimedean methods in the study of the geometry and topology of moduli spaces, with a view toward potential future applications.

The Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry.

The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT). The research station in Oaxaca is funded by CONACYT.