Let $K_0({\mathrm{Var}}_{\mathbb{Q}})[1/\mathbb{L}]$ denote the Grothendieck ring of $\mathbb{Q}$-varieties with the Lefschetz class inverted. We show that there exists a K3 surface $X$ over $\mathbb{Q}$ such that the motivic zeta function ${\zeta }_X(t):={\sum }_n[{\mathrm{Sym}}^{n}X]t^n$ regarded as an element in $K_0({\mathrm{Var}}_{\mathbb{Q}})[1/\mathbb{L}][[t]]$ is not a rational function in $t$, thus disproving a conjecture of Denef and Loeser.

We use pseudodeformation theory to study Mazur’s Eisenstein ideal. Given prime numbers $N$ and $p>3$, we study the Eisenstein part of the $p$-adic Hecke algebra for ${\Gamma }_0\left(N\right)$. We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, thereby answering a question of Mazur and generalizing a result of Calegari and Emerton. We also give new proofs of Merel’s result on this rank and of Mazur’s results on the structure of the Hecke algebra.

We prove positive-characteristic analogues of certain measure rigidity theorems in characteristic $0$. More specifically, we give a classification result for positive entropy measures on quotients of ${SL}_d$ and a classification of joinings for higher-rank actions on simply connected, absolutely almost simple groups.

In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (subcritical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field theory in the disk recently defined by Huang, Rhodes, and Vargas. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (Belavin–Polyakov–Zamolodchikov equations for degenerate field insertions) in a probabilistic setting to compute the negative moments. Finally, we will discuss applications to random matrix theory, asymptotics of the maximum of the GFF, and tail expansions of GMC.