We consider the SL(2;R) action on moduli spaces of quadratic dierentials. If is an SL(2;R)-invariant probability measure, crucial information about the associated representation on L 2 ( ) (and, in particular, ne asymptotics for decay of correlations of the diagonal action, the Te

We prove that the canonical dimension of a coadmissible representation of a semisimple p-adic Lie group in a p-adic Banach space is either zero or at least half the dimension of a nonzero coadjoint orbit. To do this we establish analogues for p-adically completed enveloping algebras of Bernstein’s inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen’s Lemma about the endomorphism ring of a simple module over an enveloping algebra.

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus ("arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is non-universal, and is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various niteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have nite rst Betti number, when the genus is at least 4. We also prove that, in this range, the I-adic completion of the Alexander invariant is nite-dimensional, and the Kahler property for the Torelli group implies the nite generation of the Johnson kernel.

These notes give a proof of the representability of homotopy invariant K-theory in the stable homotopy category of schemes (which was announced by Voevodsky). One deduces from the proper base change theorem in stable homotopy theory of schemes a descent by blow-ups theorem for homotopy invariant K-theory.

In this paper we prove that the support of a random measure on the unit ball of a separable Hilbert space that satisfies the Ghirlanda-Guerra identities must be ultrametric with probability one. This implies the Parisi ultrametricity conjecture in mean-field spin glass models, such as the Sherrington-Kirkpatrick and mixed $p$-spin models, for which Gibbs' measures are known to satisfy the Ghirlanda-Guerra identities in the thermodynamic limit.

This paper contains a proof of a conjecture of Breuil and Schneider on the existence of an invariant norm on any locally algebraic representation of GL(n), with integral central character, whose smooth part is given by a generalized Steinberg representation. In fact, we prove the analogue for any connected reductive group G. This is done by passing to a global setting, using the trace formula for an R-anisotropic model of G. The ultimate norm comes from classical p-adic modular forms.

Let F be a totally real Galois number eld. We prove the existence of base change relative to the extension F=Q for every holomorphic newform of weight at least 2 and odd level, under simple local assumptions on the eld F .

We prove the relative index conjecture, which in turn implies that the set of embeddable deformations of a strictly pseudoconvex CR-structure on a compact 3-manifold is closed in the C 1 -topology.