Abstract: We correct a mistake in the statement and proof of Lemma 2.3(d) in [Amer. J. Math. 120 (1998), no. 1, 1–21]. This in turn implies a change in Table 2.14. ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: We discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Néron-Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals' We resolve a function field variant of this problem in the large q limit, finding a connection to the z-measures first investigated in the context of harmonic analysis on the infinite symmetric group. A similar connection to z-measures is established for sums over short intervals of the divisor functions dz(n). We use these results to make conjectures in the setting of the integers which match very well with numerically produced data. Our proofs depend on equidistribution results of N. Katz and W. ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: We consider the Schrödinger evolution of strongly localized wave packets under the magnetic Laplacian in the plane ℝ2. When the initial energy is low, we obtain a precise control, in Schwartz seminorms, of the propagated states for times of order 1/ħ, where ħ is Planck's constant. In this semiclassical regime, we prove that the initial particle will always split into multiple coherent states, each one following the average dynamics of the guiding center motion but at its own speed, demonstrating a purely quantum "ubiquity" ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: Given a finite collection of C1 vector fields on a C2 manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are s+1 for s ∈ (1,∞], where s denotes the Zygmund space of order s. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the second part in a three part series of papers. The first part, joint with Stovall, addressed the same question, though the results were not sharp, and showed how such coordinate charts can be viewed as scaling maps in sub-Riemannian geometry. When viewed in ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: Suppose f ∈ L1(ℝd), Λ ⊂ ℝd is a finite union of translated lattices such that f + Λ tiles with a weight. We prove that there exists a lattice L ⊂ ℝd such that f + L also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity.Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficient criteria to determine whether a convex polygon admits translational multi-tilings. As an application, one can easily construct symmetric (2m)-gons, for any m ≥ 4, that do ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: We give a short and elementary proof of the ℓ2 decoupling inequality for the moment curve in k, using a bilinear approach inspired by the nested efficient congruencing argument of ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: Given a domain I ⊂ ℂ and an integer N > 0, a function f: I → ℂ is said to be entrywise positivity preserving on positive semidefinite N × N matrices A = (ajk) ∈ IN×N, if the entrywise application f[A] = (f(ajk)) of f to A is positive semidefinite for all such A. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic. In fixed dimension N, results akin to work of Horn and Loewner show that the first N non-zero Maclaurin coefficients of any positivity preserver f are positive; and the last N coefficients are also positive if I is unbounded. However, very little was known about the higher-order coefficients: the only examples to date for unbounded domains I were ... Read More PubDate: 2021-12-06T00:00:00-05:00

Abstract: We study the defocusing energy critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random perturbations of the initial data. The random perturbation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the perturbation.The main novelty lies in a wave packet decomposition of the random linear evolution. Through this decomposition, we can adaptively estimate the interaction between the rough and regular components of the solution. Our argument relies on techniques from restriction theory, such ... Read More PubDate: 2021-12-06T00:00:00-05:00