Spherical r-designs are Chebyshev-type averaging sets on the d-dimensional unit sphere Sd-l that are exact for all polynomials of degree at most t. The concept of such designs was introduced by Delsarte , Goethals and Seidel in 1977. The existence of spherical designs for every t and d was proved by Seymour and Zaslavsky in 1984. Although some sporadic examples are known, no general constructio...

Classical W -algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel’fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. These W -algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). Th...