A Vinnikov curve is a projective plane curve that can be written in the form det(xX + yY + zZ) = 0 for X, Y, and Z positive definite Hermitian (n × n)-matrices. Given three n-tuples of positive real numbers, α, β, and γ, there exist A, B, and C ∈ GL_nℂ with singular values α, β, and γ and ABC = 1 if and only if there is a Vinnikov curve passing through the 3n points $(-1: \alpha_i^2:0)$, $(0:-1:\beta_i^2)$, and $(\gamma_i^2:0:-1)$. Knutson and Tao proved that another equivalent condition for such A, B, and C to exist is that there is a hive (defined within) whose boundary is (log α, log β, log γ). The logarithms of the coefficients of F approximately form such a hive; this leads to a new proof of Knutson and Tao's result. This paper uses no representation theory and essentially no symplectic geometry. In their place, it uses Viro's patchworking method and a topological description of Vinnikov curves.

A locally compact group G is said to be weakly amenable if the Fourier algebra A(G) admits completely bounded approximative units. New results concerning the family of semidirect products G_n = SL(2,ℝ) $\ltimes$ H_n, n ≥ 2, together with previously known results, are used to settle the question of weak amenability for all real algebraic groups. The groups G_n fail to be weakly amenable. To show this, one follows an idea of Haagerup for the case n = 1, and one is led to the estimation of certain singular Radon transforms with product-type singularities. By representation theory, matters are reduced to a problem of obtaining rather nontrivial L²-bounds for a family of singular oscillatory integral operators in the plane, with product-type singularities and polynomial phases.

We give a formula for a cocycle generating the Hochschild cohomology of the Weyl algebra with coefficients in its dual. It is given by an integral over the configuration space of ordered points on a circle. Using this formula and a noncommutative version of formal geometry, we obtain an explicit expression for the canonical trace in deformation quantization of symplectic manifolds.

Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character is equal to a variant of the Weil distribution which occurs in André Weil's explicit formula. Hence this representation encodes information about the distribution of the prime ideals of K and is a spectral interpretation for the poles and zeros of the L-function of K. Our construction is motivated by a similar spectral interpretation by Alain Connes.