We study the Cauchy problem for the KdV equation ${\partial }_{t}u-6u{\partial }_{x}u+{\partial }_x^{3}u=0$ with almost periodic initial data $u(x,0)=V\left(x\right)$. We consider initial data $V$, for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and we show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.

The Markoff group of transformations is a group $\Gamma$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^2+y^2+z^2=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $\Gamma$ acts transitively on the set $X^{\ast }\left(p\right)$ of nonzero solutions to the same equation over $\mathbb{Z}/p\mathbb{Z}$. Recently, Bourgain, Gamburd, and Sarnak proved this conjecture for all primes outside a small exceptional set. Here, we study a group of permutations obtained by the action of $\Gamma$ on $X^{\ast }\left(p\right)$, and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $\Gamma$ acts transitively also on the set of nonzero solutions in a big class of composite moduli. Finally, our result also translates to a parallel in the case $r=2$ of a well-known theorem of Gilman and Evans regarding “$T_r$-systems” of $PSL(2,p)$.

We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than $1$, then let $r_n(\Gamma )$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (in fact, any $C>40$ suffices) such that $r_n(\Gamma )=O\left(n^C\right)$ for every such $\Gamma$. This answers a question of Larsen and Lubotzky.

We prove Sarnak’s Möbius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma \subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\in \Gamma \backslash G$ and every function $f$ continuous on the $1$-point compactification of $\Gamma \backslash G$, the sequence $f\left(x{u}^n\right)$ cannot correlate with the Möbius function on average.