If $f: Y \to X$ is a finite degree $n$ etale map of analytic (Berkovich) curves, then $X$ can have substantial regions containing points with strictly fewer than $n$ preimages. This is often caused by wild ramification, which is measured by the different, which can (Ã la Temkin et al) be thought of as a piecewise-linear function on $Y$. Understanding the slopes of the different can shed light on branched covers of algebraic curves in mixed characteristic.
Our main result is as follows: Let $S$ be an affinoid space over a $p$-adic field, parameterizing a flat family of $G$-Galois branched covers $f_s: Y_s \to {\bf P}^1$. Let $A$ be an annulus in ${\bf P}^1$. Fix a rational number $m$. For each $s$ in $S$, consider the different of $f_s$ as a function on the skeleton of $A$, identified with a closed interval $I = [0, 1]$. Under certain assumptions, the function $g: S \to I$ giving the minimum value where the different of $f_s$ has slope $\geq m$ achieves its minimum on $S$ (perhaps after a base change). This has applications to the problem of lifting branched covers of curves form characteristic $p$ to characteristic $0$, which we will discuss if time permits.
This is joint work in progress with Stefan Wewers.