We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.

We endow the set of lattices in $\mathbb{Q}_p^n$ with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake equivalence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.

Let $p$ be a maximal representation of a uniform lattice $\Gamma \subset \mathrm{SU}(n,1)$, $n\ge 2$, in a classical Lie group of Hermitian type $G$. We prove that necessarily $G=\mathrm{SU}(p,q)$ withp $p\ge qn$ there exists a holomorphic or antiholomorphic $\rho$-equivariant map from the complex hyperbolic $n$-space to thesymmetric space associated to $\mathrm{SU}(p,q)$. This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of $\mathrm{SU}(p,q)$, the representation $\rho$ extends to a representation of $\mathrm{SU}(n,1)$ in $\mathrm{SU}(p,q)$.

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. Our goal is to estimate how the stability threshold scales in $\textbf{Re}$: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data that satisfies $\Vert u_{\mathrm{in}}\Vert_{H^\sigma} \leq \delta \textbf{Re}^{-3/2}$ for any $\sigma > 9/2$ and some $\delta = \delta(\sigma) > 0$ depending only on $\sigma$ is global in time, remains within $O(\textbf{Re}^{-1/2})$ of the Couette flow in $L^2$ for all time, and converges to the class of ``2.5-dimensional" streamwise-independent solutions referred to as streaks for times $t \gtrsim \textbf{Re}^{1/3}$. Numerical experiments performed by Reddy et. al. with ``rough" initial data estimated a threshold of $\sim \textbf{Re}^{-31/20}$, which shows very close agreement with our estimate.

This paper is a continuation and a completion of the work of the first and the third author on the Jordan decomposition. We extend the Jordan decompositionof blocks: we show that blocks of finite groups of Lie type innondescribing characteristic are Morita equivalent to blocks of subgroupsassociated to isolated elements of the dual group --- this is the modular version of a fundamental result of Lusztig, and the best approximation of the character-theoretic Jordan decomposition that can be obtained via Deligne-Lusztig varieties. The key new result is theinvariance of the part of the cohomology in a given modular seriesof Deligne-Lusztig varieties associated to a given Levi subgroup,under certain variations of parabolic subgroups.

We also bring in local block theory methods: we show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of the original work of the first and the third author, the finer homotopy equivalencewas unknown. As a consequence, the equivalences preserve defect groupsand categories of subpairs. We finally determine when Deligne-Lusztiginduced representations of tori generate the derived category of representations. An additional new feature is an extension of the results to disconnected reductive algebraic groups, which is required to handle local subgroups.

Laczkovich proved thatif bounded subsets $A$ and $B$ of $\mathbb{R}^k$ have the same nonzero Lebesgue measure and the upper box dimension of the boundary of each set is less than $k$, then there is a partitionof $A$ into finitely many parts that can be translated to form a partitionof $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaringand Hilbert's third problem.