In the first part of my talk, I will introduce various flavours of tropicalization, a process that associates piecewise-linear polyhedral objects to algebraic varieties. I will explain how this piecewise linear geometry ties intimately to degeneration and compactification of (families of) algebraic varieties. Nonarchimedean analytic geometry, in the sense of Berkovich, provides an elegant and powerful framework in which to study these so-called skeletons.
In the second part of my talk we will explore the applications of tropicalization, as a degeneration technique, to computing enumerative invariants. I will focus on the case of target curves, where insights from the theory of Berkovich spaces have begun to dissolve some of the mystery behind the success of these combinatorial techniques. For instance, the relationship between the analytification of Hurwitz space and its skeleton precisely explains how and why tropical geometry can be used to compute enumerative invariants of target curves.
For a sub-additive ergodic sequence {X_{m,n}}, Kingman's theorem gives convergence for the terms X_{0,n}/n to some non-random number g. In this talk, I will discuss the convergence rate of the mean EX_{0,n}/n to g. This rate turns out to be related to the size of the random fluctuations of X_{0,n}; that is, the variance of X_{0,n}, and the main theorems I will present give a lower bound on the convergence rate in terms of a variance exponent (if it exists). The main assumptions are that the sequence is not diffusive (the variance does not grow linearly) and that it has a weak dependence structure. Various examples, including first and last passage percolation, bin packing, and longest common subsequence fall into this class. This is joint work with Tuca Auffinger and Jack Hanson.
Rate of convergence of the mean for sub-additive ergodic sequences
