We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions  have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We use proof-nets to study the algorithmic complexity of the derivability problem for some fragments of the Lambek calculus. We prove the NP-completeness of this problem for the unidirectional fragment and the product-free fragment, and also for versions of these fragments that admit empty antecedents.

We develop a theory of direct families of polytopes with moment-angle-manifolds having nontrivial strictly defined Massey products of arbitrarily large order in cohomology.

The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces G/H, where G is a simply connected non-simple semisimple complex algebraic group and H is a connected closed subgroup of G. In all cases we also find the highest-weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups.