We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the straightening algorithm.

In this paper we prove a conjecture of Jacquet about supercuspidal representations of GL_n(K) distinguished by GL_n(k), or by U_n(k), for K a quadratic unramified extension of a non-Archimedean local field k.

Let X be a p-divisible group over a regular scheme S such that the Newton polygon in each geometric point of S is the same. Then there is a p-divisible group isogenous to X which has a slope filtration.

A ℚ-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every ℚ-curve is modular, and he showed that a positive answer would follow from J.-P. Serre's conjecture on mod p Galois representations. We answer Ribet's question in the affirmative, subject to certain local conditions at 3.

An analogue of the Fourier transform is developed for D-algebras. G. Laumon's equivalence between the derived category of $\mathscr {D}$-modules on an abelian variety and the derived category of $\mathscr {O}$-modules on the universal extension of the dual variety is seen as a degenerate case of a duality for twisted differential operators (tdo's) with respect to which the dual of a generic tdo is again a tdo.

By looking at the average behavior (n-level density) of the low-lying zeros of certain families of L-functions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups.

On the space of nondepraved (see [8]) real, isolated singularities, we consider the stable equivalence relation induced by smooth deformations whose asymptotic behaviour is controlled by the Palais-Smale condition. It is shown that the resulting space of equivalence classes admits a canonical semiring structure and is isomorphic to the semiring of stable homotopy classes of CW-complexes.

In an application to Hamiltonian dynamics, we relate the existence of bounded and periodic orbits on noncompact level hypersurfaces of Palais-Smale Hamiltonians with just one singularity that is nondepraved to the lack of self-duality (in the sense of E. Spanier and J. Whitehead) of the sublink of the singularity.