Georgia Institute of Technology College of Sciences

We will discuss the relationship between a Berkovich analytic curve over a complete and algebraically closed non-Archimedean field and its tropicalizations, paying special attention to the natural metric structure on both sides. This is joint work with Sam Payne and Joe Rabinoff.

Tropicalization is a procedure for associating a polyhedral complex to a subvariety of an algebraic torus. We study the question on which graphs arise from tropicalizing algebraic curves. By using Baker's technique of specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition reproduces the known necessary conditions and also gives new conditions.

I will discuss the correspondence between non-archimedean amoebas and tropical varieties, which is a generalization of the theory of Newton polygons to polynomials in several variables.

The moduli space of representations of a fundamental group of a knot in SL(2,C) is an affine algebraic variety, and generically a complex curve, with an explicit projection to C^2. The ideal that defines this curve has special type described by binomial and linear equations. I will motivate this curve using elementary hyperbolic geometry, and its Newton polygon in the plane using geometric topology. Finally, I will describe a heuristic method for computing the Newton polygon without computing the curve itself, using tropical implitization, work in progress with Josephine Yu. The talk will be concrete, with examples of concrete curves that come from knots. This talk involves classical mathematics. A sequel of it will discuss quantum character varieties of knots and tropical curves.