Let$S$be a closed Shimura variety uniformized by the complex$n$-ball associated with a standard unitary group. The Hodge conjecture predicts that every Hodge class in $${H^{2k} (S, \mathbb{Q})}$$ , $${k=0,\dots, n}$$ , is algebraic. We show that this holds for all degrees$k$away from the neighborhood $${\bigl]\tfrac13n,\tfrac23n\bigr[}$$ of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension$c$of the subvariety) centered at the middle dimension of S. These results are derived from a general theorem that applies to all Shimura varieties associated with standard unitary groups of any signature. The proofs make use of Arthur’s endoscopic classification of automorphic representations of classical groups. As such our results rely on the stabilization of the trace formula for the (disconnected) groups $${GL (N) \rtimes \langle \theta \rangle}$$ associated with base change.

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years.

We establish a parametric extension$h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the 3-dimensional result from [12]. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures.

Let$f$be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, $${f \in \mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ , $${\# A< \infty}$$ . J. Nuttall has put forward the important relation between the$maximal domain$of$f$where the function has a single-valued branch and the$domain of convergence$of the diagonal Padé approximants for$f$. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of$f$in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function$f$single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has$minimal logarithmic capacity$among all other systems converting the function$f$to a single-valued branch. Thus the domain of convergence corresponds to the$maximal$(in the sense of$minimal$boundary) domain of single-valued holomorphy for the analytic function $${f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ . The complete proof of Nuttall’s conjecture (even in a more general setting where the set$A$has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive$strong asymptotics$for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that$A$is a finite set of branch points of$f$which have the$algebro-logarithmic character$and which are placed in a$generic position$. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

We prove the Mirković–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no$p$-torsion, as long as$p$is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when$p$is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.