Volume 2016, Issue 712

We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for the sum of eigenvalues with the leading (Weyl) term given by the volume and the subleading term by the surface area. Our result is valid under very weak assumptions on the regularity of the boundary.

In this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of ℚ. Using the projective metric studied in [Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 2, 211–248] we prove the analogue of Khintchine's theorem in projective space. For finite places and in higher dimension, we are able to completely remove the condition of monotonicity and establish the analogue of the Duffin–Schaeffer conjecture.

We study a quasilinear parabolic equation of forward-backward type, under assumptions on the nonlinearity which hold for a wide class of mathematical models, using a pseudo-parabolic regularization of power type. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. It is shown that these solutions satisfy suitable entropy inequalities. We also study their qualitative properties, in particular proving that the singular part of the solution with respect to the Lebesgue measure is constant in time.

We classify isolated singularities f∈K[[x1,...,xn]]${f\in K[[x_1,\ldots , x_n]]}$, which are simple, i.e. have no moduli, with respect to right equivalence, where K is an algebraically closed field of characteristic p > 0. For K=ℝ${K=\mathbb {R}}$ or ℂ this classification was initiated by Arnol'd, resulting in the famous ADE-series. The classification with respect to contact equivalence for p > 0 was done by Greuel and Kröning with a result similar to Arnol'd's. It is surprising that with respect to right equivalence and any given p > 0 we have only finitely many simple singularities, i.e. there are only finitely many k such that A k and D k are right simple, all the others have moduli. We conjecture a similar finiteness result for singularities with an arbitrary number of moduli. A major point of this paper is the generalisation of the notion of modality to the algebraic setting, its behaviour under morphisms, and its relations to formal deformation theory. As an application we show that the modality is semicontinuous in any characteristic.

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties.

We investigate simple endotrivial modules of finite quasi-simple groups and classify them in several important cases. This is motivated by a recent result of Robinson [Bull. Lond. Math. Soc. 43 (2011), 712–716] showing that simple endotrivial modules of most groups come from quasi-simple groups.

We use an idea of Rosenberg to prove a reconstruction theorem for abelian categories of α-twisted quasi-coherent sheaves on quasi-compact and quasi-separated schemes X when α∈Br(X)$\alpha \in \operatorname{Br}(X)$. By applying the work of Toën on derived Azumaya algebras, we give a proof of Căldăraru's conjecture.

In this paper we define the notion of a hyperkähler manifold (potentially) of Jacobian type. If we view hyperkähler manifolds as “abelian varieties”, then those of Jacobian type should be viewed as “Jacobian varieties”. Under a minor assumption on the polarization, we show that a very general polarized hyperkähler fourfold F of K3[2]${K3^{[2]}}$-type is not of Jacobian type. As a potential application, we conjecture that if a cubic fourfold is rational then its variety of lines is of Jacobian type. Under some technical assumption, it is proved that the variety of lines on a rational cubic fourfold is potentially of Jacobian type. We also prove the Hodge conjecture in degree 4 for a generic F of K3[2]${K3^{[2]}}$-type.

Let X be a finite set and let k be a commutative ring. We consider the k -algebra of the monoid of all relations on X , modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card( X ), called inessential relations. This quotient is called the essential algebra associated to X . We then define a suitable nilpotent ideal of the essential algebra and describe completely the structure of the corresponding quotient, a product of matrix algebras over suitable group algebras. In particular, we obtain a description of all the simple modules for the essential algebra.