We reduce the computation of Poisson traces on quotients of symplectic vector
spaces by finite subgroups of symplectic automorphisms to a finite one, by
proving several results which bound the degrees of such traces as well as the
dimension in each degree. This applies more generally to traces on all
polynomial functions which are invariant under invariant Hamiltonian flow.

We develop representation theory of the rational Cherednik algebra H
associated to a finite Coxeter group W in a vector space h. It is applied to
show that, for integral values of parameter `c', the algebra H is simple and
Morita equivalent to D(h)#W, the cross product of W with the algebra of
polynomial differential operators on h.

We determine the support of the irreducible spherical representation (i.e.,
the irreducible quotient of the polynomial representation) of the rational
Cherednik algebra of a finite Coxeter group for any value of the parameter c.
In particular, we determine for which values of c this representation is finite
dimensional. This generalizes a result of Varagnolo and Vasserot,
arXiv:0705.2691, who classified finite dimensional spherical representations in
the case of Weyl groups and equal parameters (i.e., when c is a constant
function).

We apply the yoga of classical homotopy theory to classification problems of
G-extensions of fusion and braided fusion categories, where G is a finite
group. Namely, we reduce such problems to classification (up to homotopy) of
maps from BG to classifiying spaces of certain higher groupoids. In particular,
to every fusion category C we attach the 3-groupoid BrPic(C) of invertible
C-bimodule categories, called the Brauer-Picard groupoid of C, such that
equivalence classes of G-extensions of C are in bijection with homotopy classes
of maps from BG to the classifying space of BrPic(C).