This work is a theoretical study of Toeplitz operators the symbol of which is a regular matricial function, positive definite everywhere on the one-dimensionnal torus. We propose a trace theorem with an asymptotic expression giving an extension of the three Szegö limit theorems to the matricial case.

We consider the initial value problem for systems of cubic nonlinear Schrödinger equations in one space dimension with small initial data. We present a structural condition on the nonlinearity under which the solution exists globally in time and behaves like a free solution at infinity. This condition corresponds to an NLS version of the \emph{null condition}.

In this paper, it is proved that if, at each point of a Weyl manifold, the sectional curvature is independent of the plane chosen, then the Weyl manifold is locally conformal to an Einstein manifold and that the scalar curvature of the Weyl manifold is prolonged covariant constant.

We characterize the weighted weak type inequalities with variable exponents for the modified Hardy operators and the Hardy-Littlewood maximal operators.

The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki, asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra $\mathfrak{g}$ where $\mathfrak{g}$ is $\mathfrak{gl}_\infty$ or the affine Lie algebra $A^{(1)}_\ell$, and the irreducible representations correspond to the upper global bases. In this note, we formulate analogous conjectures for certain classes of irreducible representations of affine Hecke algebras of type B.

It is shown that any elliptic K3 surface with a section and with two $II^*$-fibres is sandwiched by a Kummer surface in a very precise way.

We show that the notion of zeta functions over the field of one element $\F_1$, as given in special cases by Soulé, extends naturally to all $\F_1$-schemes as defined by the author in an earlier paper. We further give two constructions of K-theory for affine schemes or $\F_1$-rings, we show that these coincide in the group case, but not in general.