In a scale of Fock spaces with radial weights ϕ we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if ϕ(x) grows at most like (log x)2.

We introduce certain energy functionals to complex Monge–Ampère equations over bounded domains with inhomogeneous boundary conditions, and use these functionals to show the convergence of solutions to certain parabolic Monge–Ampère equations.

A theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K(Ator) is Hilbertian. We prove our conjecture when K is a number field. The proof applies a result of Serre about l-torsion of Abelian varieties, information about l-adic analytic groups, and Haran's diamond theorem.

We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing of SK2(A) for a central simple algebra A of square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg–Chekanov twist knots En are not transversely simple for n odd and n > 3.

Since any function f(x1, … , xm) from {0, 1}m in a finite field k can be uniquely written as a multilinear polynomial, we associate to it its inverse dual f*(x1, … , xm) expressing the coefficients of this canonical polynomial. We show that the unlikely hypothesis that the class P(k) of sequences of functions of polynomial complexity be closed by duality is equivalent to the well-known hypothesis P = #pP, where p is the characteristic of k.

In a first section we expose the result in the frame of classical Boolean calculus, that is, when k = ℤ/2ℤ. In a second section we treat the general case, introducing a notion of transformation whose duality is a special case; the transformations form a group isomorphic to GL2(k); among them, we distinguish the benign transformations, which have a weak effect on the complexity of functions; we show that, in this respect, all the non-benign transformations have the same power of harmfulness.

In the third section we consider functions from km into k, and in the last, after introducing #P = P to the landscape, we compare our results with those of Guillaume Malod, concerning the closure by ‘coefficient-function’ of various classes of complexity of sequences of polynomial defined in Valiant's way.

A factorization formula for certain automorphisms of a Poisson algebra associated with a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulae for Donaldson–Thomas type invariants of Kontsevich and Soibelman.

The statement of Theorem 2.7.6 in the indicated paper is incorrect. For instance, if m = 0 (i.e., the base field K contains no additional derivations), it is inconsistent with Theorem 2.7.4 due to the distinction between intrinsic and extrinsic generic radii of convergence. Theorems 2.7.12 and 2.7.13 are incorrect for similar reasons.