Volume 21, Issue 5

This article uses techniques from multivariate asymptotic analysis to prove a result of Ikromov-Kiehl-Müller that approximates those p for which a certain maximal operator associated to the graph in of a binary form is bounded on .

A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to S 2 is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity at the origin in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controlled Kepler system are finally obtained thanks to the computation of the cut locus of the restriction to the sphere.

We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

The pure braid group Γ of a quadruply-punctured Riemann sphere acts on the SL(2, ℂ)-moduli ℳ of the representation variety of such sphere. The points in ℳ are classified into Γ -orbits. We show that, in this case, the monodromy groups of many explicit solutions to the Riemann-Hilbert problem are subgroups of SU(2). Most of these solutions are examples of representations that have dense images in SU(2), but with finite Γ -orbits in ℳ. These examples relate to explicit immersions of constant mean curvature surfaces.

We show how eigentheory clarifies many algebraic properties of Cayley-Dickson algebras. These notes are intended as background material for those who are studying this eigentheory more closely.

The Cayley-Dickson algebras A n are an infinite sequence of (non-associative) algebras beginning with the well-known composition algebras ℝ, ℂ, ℍ, 𝕆. We completely describe all possible dimensions for the alternator Alt( a ) ≔ { b ∈ A n | a ( ab ) = ( aa ) b = 0} of an element a ∈ A n , for n ≥ 7. This resolves a conjecture of Biss, Christensen, Dugger, and Isaksen. On the way to obtaining this result, we establish numerous results on the eigentheory of left multiplication operators in A n , some of which may be of independent interest.

A vector bundle E on a smooth irreducible algebraic variety X is called a Steiner bundle of type ( F 0 , F 1 ) if it is defined by an exact sequence of the form where s, t ≥ 1 and ( F 0 , F 1 ) is a strongly exceptional pair of vector bundles on X such that is generated by global sections. Let X be a smooth irreducible projective variety of dimension n with an n -block collection , of locally free sheaves on X which generate D b (𝒪 X – mod ). We give a cohomological characterisation of Steiner bundles of type on X , where 0 ≤ a < b ≤ n and 1 ≤ i 0 ≤ α a , 1 ≤ j 0 ≤ α b .

We give a characterization of Σ -cotorsion modules over valuation domains in terms of descending chain conditions on certain chains of definable subgroups. We prove that pure submodules, direct products and modules elementarily equivalent to a Σ -cotorsion module are again Σ -cotorsion. Moreover, we describe the structure of Σ -cotorsion modules.

To any connected and simply connected nilpotent Lie group N , one can associate its group of affine transformations Aff ( N ). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N , via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim( G ) = dim( N ) ≤ 5. Finally, we specialize to the case of abelian simply transitive affine actions on a given connected and simply connected nilpotent Lie group. It turns out that such a simply transitive abelian affine action on N corresponds to a particular Lie compatible bilinear product on the Lie algebra 𝔫 of N , which we call an LR-structure.

We prove that the moduli space ℳ g,n of smooth curves of genus g with n marked points is rational for g = 6 and 1 ≤ n ≤ 8, and it is unirational for g = 8 and 1 ≤ n ≤ 11, g = 10 and 1 ≤ n ≤ 3, g = 12 and n = 1.