Berkovich Skeleta and Wildly Ramified Curves

In mathematics, the field of arithmetic geometry is essentially the study of arithmetic problems like solving equations over the rational numbers using methods inspired from geometry. An equation like x²+y²=1 defines a curve in the plane, in this case a circle, and finding rational solutions of this equation, like (x,y)=(3/5,4/5), is the same as finding special points on this circle, the so-called rational points. Finding rational points on an arithmetic curve is a deep problem, it was for example only proved in the 90's by Wiles and Taylor that all the rational solutions to x^n+y^n=1 for n larger than 2 are (1,0),(-1,0), (0,1) and (0,-1); this is famously known as Fermat's last theorem. A succesful method for finding rational points is through understanding the so-called reduction properties of the curve. For example, we might look at an equation like x²=3y²-1 and reduce it modulo 3 (divide by 3 with residue), to see there are no solutions. A good understanding of the reductions of the curve modulo the different primes and 'glueing' these together can give us insight in the global phenomena. This project is studies especially ill-behaved reductions, the so-called wildly ramified case, in which many interesting problems remain open, like finding appropriate ways to measure how ill-behaved the curve is. We will attempt answerring some of these questions by applying geometric techniques due to Berkovich, providing a unique interplay between analysis, geometry and algebra.