We prove a 1999 conjecture of Veys, which says that the opposite of the log-canonical threshold is the only possible pole of maximal order of Denef and Loeser’s motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic $0$. We apply similar methods to study the weight function on the Berkovich skeleton associated with a degeneration of Calabi–Yau varieties. Our results suggest that the weight function induces a flow on the nonarchimedean analytification of the degeneration towards the Kontsevich–Soibelman skeleton.

We show that the $p$-adic eigencurve is smooth at classical weight $1$ points which are regular at $p$ and give a precise criterion for étaleness over the weight space at those points. Our approach uses deformations of Galois representations.

We prove that for any pair $(s,t)$ of nonnegative numbers with $s+t=1$, the set of $2$-dimensional $(s,t)$-badly approximable vectors is winning for Schmidt’s game. As a consequence, we give a direct proof of Schmidt’s conjecture using his game.

We introduce two new general methods to compute the Chow motives of twisted flag varieties and settle a 20-year-old conjecture of Markus Rost about the Rost invariant for groups of type ${\mathrm{E}}_7$.

Let $\mu$ be a probability measure on $Out\left(F_N\right)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $Out\left(F_N\right)$. We show that almost every sample path of the random walk on $(Out(F_N),\mu )$, when realized in Culler and Vogtmann’s outer space, converges to the simplex of a free, arational tree. We then prove that the space $\mathcal{FI}$ of simplices of free and arational trees, equipped with the hitting measure, is the Poisson boundary of $(Out(F_N),\mu )$. Using Bestvina and Reynolds’s and Hamenstädt’s description of the Gromov boundary of the complex ${\mathcal{FF}}_N$ of free factors of $F_N$, this gives a new proof of the fact, due to Calegari and Maher, that the realization in ${\mathcal{FF}}_{\mathcal{N}}$ of almost every sample path of the random walk converges to a boundary point. We get in addition that $\partial {\mathcal{FF}}_N$, equipped with the hitting measure, is the Poisson boundary of $(Out(F_N),\mu )$.

We explore the dynamics of the action of the mapping class group in genus $2$ on the ${PSL}_2\left(\mathbb{R}\right)$-character variety. We prove that this action is ergodic on the connected components of Euler class $\pm 1$, as it was conjectured by Goldman. In the connected component of Euler class $0$ there are two invariant open subsets; on one of them the action is ergodic. In this process we give a partial answer to a question posed by Bowditch.