Powersum varieties, also called varieties of sums of powers, have provided examples of interesting relations between varieties since their first appearance in the 19th century. One of the most useful tools to study them is apolarity, a notion originally related to the action of differential operators on the polynomial ring. In this work, we make explicit how one can see apolarity in terms of the Cox ring of a variety. In this way, powersum varieties are a special case of varieties of apolar schemes; we explicitly describe examples of such varieties in the case of two toric surfaces, when the Cox ring is particularly well-behaved.

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck’s Torus Theorem, and obtain corresponding results for p-compact groups and p-Noetherian groups.

We prove that almost all random subsets of a finite vector space are weak Salem sets (small Fourier coefficient), which extend a result of Hayes to a different probability model.

We study analytic integrable deformations of the germ of a holomorphic foliation given by df=0 at the origin 0∈Cn,n≥3. We consider the case where f is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, outside of a dimension ≤n−3 analytic subset Y⊂X, the analytic hypersurface Xf:(f=0) has only normal crossings singularities. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of integrable germs writing as ω=df+fη where f is quasi-homogeneous. Under the same hypotheses for Xf:(f=0) we prove that ω also admits a holomorphic first integral. Finally, we conclude that an integrable germ ω=adf+fη admits a holomorphic first integral provided that: (i) Xf:(f=0) is irreducible with an isolated singularity at the origin 0∈Cn,n≥3; (ii) the algebraic multiplicities of ω and f at the origin satisfy ν(ω)=ν(df). In the case of an isolated singularity for (f=0) the writing ω=adf+fη is always assured so that we conclude the existence of a holomorphic first integral. Some questions related to Relative Cohomology are naturally considered and not all of them answered.

We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings, it is proved that these are exactly the torsion T such that the regular module has a special T-preenvelope. In particular, every torsion-enveloping class in Mod-R are of the form Gen(T) for a minimal silting module T. For the dual case, we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form Cogen(T), where T is a cosilting module.