T&G: Jeff Hicks (Cambridge), Realizability and Obstruction in Tropical Geometry

Tue, 08/06/202118:00-19:00
Tropicalization is a process that inputs a piece of algebraic geometry and outputs a tropical --- piecewise-linear --- object. A major problem in the field is understanding when this process is reversible. For example, Mikhalkin showed that every tropical curve in the plane is the tropicalization of some algebraic curve. This reverse process is called ``realization''. Since there are known examples of tropical curves in 3-dimensional space which are not the tropicalization of any algebraic curve, the dictionary between tropical and algebraic geometry is imperfect. Having realization criteria allows us to attack difficult problems in enumerative geometry via combinatorics in the tropical world.

In this talk, we will examine a similar phenomenon in symplectic geometry. We will discuss the symplectic realization problem, which asks when there exists an unobstructed Lagrangian lift of a tropical curve whose moment map image approximates the tropical data. Additionally, we will explain why in certain cases unobstructedness implies realizability, and see how realization criteria may be recovered from symplectic geometry.