I will introduce the concept of Berkovich analytification and discuss briefly the motivations that led to its definition. We will go through the interesting example of the affine line, state the inverse limit theorem proved by Sam Payne, linking the tropical and the Berkovich world, and sketch its proof.
For more information, please check: sachihashimoto.github.io/workshop
Let C:y^2=f(x) be a hyperelliptic curve over a local field of odd residue characteristic. I will discuss joint work with Tim and Vladimir Dokchitser, and Celine Maistret, demonstrating how several arithmetic invariants of the curve and its Jacobian, most notably its potential stable reduction, can be described in terms of simple combinatorial data involving the p-adic distances between the roots of f(x).
For more information, please check: sachihashimoto.github.io/workshop
Tropical geometry is a combinatorial shadow of classical geometry. Algebraic curves in the tropical plane are dual to triangulations of convex polygons. We discuss the intrinsic geometry of these objects, with focus on the moduli space of metric graphs that represent tropical plane curves. This lectures is based on work with Sarah Brodsky, Michael Joswig and Ralph Morrision from nearly a decade ago (arXiv:1409.4395).
For more information, please check: sachihashimoto.github.io/workshop
This short talk is meant to advertise a seminar to be hold this spring/summer about tropical aspects of mirror symmetry. We will mainly be interested in the case of projective spaces (or even only P^2), and follow the book of Mark Gross "Tropical geometry and mirror symmetry". This talk will briefly review classical mirror symmetry of P^2 (expressed as an isomorphism of certain systems of differential equations), and then comment on the program for the seminar.
For more information, please check: sachihashimoto.github.io/workshop
I will explain how to describe models of curves using valuations. This will be applied to studying models of the projective line and models of coverings of curves. Specifically, we will discuss the problem of determining the semistable reduction of plane quartics via suitable maps to the projective line, including the wild case of reduction at p=3.
For more information, please check: sachihashimoto.github.io/workshop
In this talk, I will give a general technique to find poset substructures of schemes using discrete Galois-data associated to coverings. In particular, this gives a fast algorithm to calculate dual intersection complexes of semistable models of varieties, provided a suitable covering is given. I will show how tropical geometry and schön compactifications of branch loci can be used to generate these coverings.
For more information, please check: sachihashimoto.github.io/workshop
In this introductory lecture, we will study curves over valuated fields and their combinatorial structure. Tropical geometry and number theory study the degeneration of curves in this setting. We give definitions from both fields, discuss semistable models, and give motivation for the following week's lectures.
For more information, please check: https://sachihashimoto.github.io/workshop
