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An irreducible integrable connection (E,∇) on a smooth projective complex variety X is called rigid if it gives rise to an isolated point of the corresponding moduli space M_{dR}(X). According to Simpson’s motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauss-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of X. In this article we study mod-p reductions of irreducible rigid connections and establish results which confirm Simpson’s prediction. In particular, for large p, we prove that p-curvatures of mod-p reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an F-isocrystalline realization for irreducible rigid flat connections. More precisely, we prove that there exist smooth models X_R and (E_R,∇_R) of X and (E,∇), over a finite-type ring R, such that for every Witt ring W(k) of a finite field k and every homomorphism R→W(k), the p-adic completion of the base change (\hat E_{W(k)}, \hat ∇_{W(k)}) on \hat X_{W(k)} represents an F-isocrystal. Subsequently, we show that irreducible rigid flat connections with vanishing p-curvatures are unitary. This allows us to prove new cases of the Grothendieck–Katz p-curvature conjecture. We also prove the existence of a complete companion correspondence for F-isocrystals stemming from irreducible cohomologically rigid connections.

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs—in contrast, in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.

We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.

The main results of this paper involve general algebraic differentials ω on a general pencil of algebraic curves. We show how to determine whether ω is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck–Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin–Mumford type allied to the Zilber–Pink conjectures: we characterise torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations A^2−DB^2=1 over a polynomial ring. We determine whether the Pell equation (with squarefree D) is solvable for infinitely many members of the pencil.

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most 1, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston’s conjecture. Here counter-examples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.

In his seminal 1976 paper, Bill Thurston observed that a closed leaf S of a codimension‑1 foliation on a compact 3‑manifold has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation F′ such that S is homologous to a union of leaves and such that the plane field of F′ is homotopic to that of F. In particular, F and F′ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3‑manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston’s conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.

We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan-Gross-Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding L-functions. Ratner’s results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher-rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino-Ikeda conjecture.

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P^n, in which case Bir(X) is the Cremona group of rank n, or when X⊂P^{n+1} is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank n⩾3 is not generated by linear and Jonquières elements.

Using the geometric Satake correspondence, the Mirković–Vilonen cycles in the affine Grasssmannian give bases for representations of a semisimple group G. We prove that these bases are “perfect”, i.e. compatible with the action of the Chevelley generators of the positive half of the Lie algebra g. We compute this action in terms of intersection multiplicities in the affine Grassmannian. We prove that these bases stitch together to a basis for the algebra C[N] of regular functions on the unipotent subgroup. We compute the multiplication in this MV basis using intersection multiplicities in the Beilinson–Drinfeld Grassmannian, thus proving a conjecture of Anderson. In the third part of the paper, we define a map from C[N] to a convolution algebra of measures on the dual of the Cartan subalgebra of g. We characterize this map using the universal centralizer space of G. We prove that the measure associated with an MV basis element equals the Duistermaat–Heckman measure of the corresponding MV cycle. This leads to a proof of a conjecture of Muthiah. Finally, we use the map to measures to compare the MV basis and Lusztig’s dual semicanonical basis. We formulate conjectures relating the algebraic invariants of preprojective algebra modules (which underlie the dual semicanonical basis) and geometric invariants of MV cycles. In the appendix, we use these ideas to prove that the MV basis and the dual semicanonical basis do not coincide in SL_6 .

We show that there are 2^{2^{ℵ_0}} different closed ideals in the Banach algebra L(L_p(0,1)),1<p≠2<∞. This solves a problem in A. Pietsch’s 1978 book “Operator Ideals”. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non-Hilbertian. We give a criterion for a space with an unconditional basis to have 2^{2^{ℵ_0}} closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for 1<q<2 the space Xq of Rosenthal, which is isomorphic to a complemented subspace of L_q(0,1), admits such an operator.