We obtain a representation of the derivative of the spectral shift function $\xi(\lambda,h)$ in the framework of semiclassical "black box" perturbations. Our representation implies a meromorphic continuation of $\xi(\lambda,h)$ involving the semiclassical resonances. Moreover, we obtain a Weyl-type asymptotics of the spectral shift function, as well as a Breit-Wigner approximation in an interval $(\lambda -\delta,\lambda+\delta), 0<\delta<\epsilon h$.

We compute the dimension of the tangent space to, and the Krull dimension of, the prorepresentable hull of two deformation functors. The first one is the "algebraic" deformation functor of an ordinary curve $X$ over a field of positive characteristic with prescribed action of a finite group $G$, and the data are computed in terms of the ramification behaviour of $X\to G\backslash X$. The second one is the "analytic" deformation functor of a fixed embedding of a finitely generated discrete group $N$ in ${\rm PGL}(2, K)$ over a nonarchimedean-valued field $K$, and the data are computed in terms of the Bass-Serre representation of $N$ via a graph of groups. Finally, if $\Gamma$ is a free subgroup of $N$ such that $N$ is contained in the normalizer of $\Gamma$ in ${\rm PGL}(2, K)$, then the Mumford curve associated to $\Gamma$ becomes equipped with an action of $N/\Gamma$, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of $N$.

We show that there exist mixing zero-entropy algebraic $\mathbb {Z}\sp 8$-actions that are measurably and topologically conjugate but not algebraically conjugate. This result gives the first known examples of mixing zero-entropy algebraic $\mathbb {Z}\sp d$-actions that do not have rigidity properties, and it provides a negative answer to the isomorphism rigidity problem raised in B. Kitchens and K. Schmidt (Isomorphism rigidity of irreducible algebraic $\mathbb {Z}\sp d$-actions, Invent. Math. 142 (2000), 559‒577) and Schmidt ("The dynamics of algebraic $\mathbb {Z}\sp d$-actions" in European Congress of Mathematics (Barcelona, 2000), Vol. 1, Progr. Math. 201, Birkhäuser, Basel, 2001, 543‒553).

Let $\mathfrak {g}$ be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of $\mathfrak {g}$ are described: tensor product and multiplicity varieties. These varieties are closely related to Nakajima's quiver varieties and should play an important role in the geometric constructions of tensor products and intertwining operators. In particular, it is shown that the set of irreducible components of a tensor product variety can be equipped with the structure of a $\mathfrak {g}$-crystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finitedimensional representations of $\mathfrak {g}$, and that the number of irreducible components of a multiplicity variety is equal to the multiplicity of a certain representation in the tensor product of several others. Moreover, the decomposition of a tensor product into a direct sum is described geometrically.

Hexagonal circle patterns with constant intersection angles are introduced and studied. It is shown that they are described by discrete integrable systems of Toda type. Conformally symmetric patterns are classified. Circle pattern analogs of holomorphic mappings $z\sp c$ and $\log z$ are constructed as special isomonodromic solutions. Circle patterns studied in the paper include Schramm's circle patterns with the combinatorics of the square grid as a special case.

We define a collection of polytopes associated to a semisimple group $\mathsf {G}$. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain region. The polytopes are defined as moment map images of algebraic cycles discovered by I. Mirković and K. Vilonen. These cycles are a canonical basis for the intersection homology of (the closures of the strata of) the loop Grassmannian.