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This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of observing any diffeomorphism type and any nesting arrangement among the zero set components is strictly positive for waves of large enough frequencies. Our results are a consequence of building Laplace eigenfunctions in euclidean space whose zero sets have a component with prescribed topological type or an arrangement of components with prescribed nesting configuration. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band‐limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular, the results apply to random monochromatic waves and to random real algebraic hypersurfaces in projective space. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

MR2415383 AN APPRECIATION OF JEAN BOURGAIN'S WORK 153 [BG08b] J. Bourgain and A. Gamburd, On the spectral gap for finitely-generated subgroups of SU(2), Invent. The four articles in this issue give very clear and insightful accounts of some highlights of Jean's work in functional analysis (Ball), harmonic analysis (Demeter), dispersive partial differential equations (Kenig), and a deconstruction of some of the techniques that Jean invented (Tao). BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 58, Number 2, April 2021, Pages 151-153 https://doi.org/10.1090/bull/1732 Article electronically published on February 12, 2021 AN APPRECIATION OF JEAN BOURGAIN'S WORK PETER SARNAK Jean Bourgain viewed himself as an "analyst", and as the record shows he was uniquely gifted as such, and much more. [Extracted from the article]

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature .n - 1; 1/is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for nFn-1 are thin. [ABSTRACT FROM AUTHOR]

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity. [ABSTRACT FROM AUTHOR]

We report on some recent striking advances on the quantum unique ergodicity, or QUE conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved manifold. The account falls naturally into two categories. The first concerns the general conjecture where the tools are more or less limited to microlocal analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. Our main emphasis is on the second category, especially where QUE has been proven. This note is not meant to be a survey of these topics, and the discussion is not chronological. Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context. [ABSTRACT FROM AUTHOR]

The article focuses on the creation of integral Apollonian circle packings, which feature Diophantine problems. It states that American coins are placed for the circle packings which include quarter, nickel, and dime. It mentions a theorem which reveals that positive density in all the positive integers are present in the set of curvatures in an integral Apollonian packing. It adds that primitive integral Apollonian packing has infinitely lots of twin primes and circles with prime curvatures.

The existence of a strong spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg conjectures. If G has no compact factors then for general lattices a spectral gap can still be established, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice Γ in G = PSL(2, R)d for d ⩾ 2, which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting. [ABSTRACT FROM AUTHOR]

The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d – 1 is a prime power. In this paper, we consider the case when d – 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost” Ramanujan graphs for almost every value of d. More precisely, for any fixed ℇ > 0 and for almost every value of d (in the sense of natural density), there are infinitely many d-regular graphs such that all the non-trivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ℇ) . 2000 Mathematics Subject Classification: 05C50, 15A18, 11N05, 11F30. [ABSTRACT FROM AUTHOR]

For a fixed hyperelliptic curve C given by the equation y2 = f( x) with f ∈ ℤ[ x] having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm whose equation is given by my2 = f( x). More precisely, we study the Dirichlet series where the summation is over all non-zero squarefree integers. We show that converges for ℜ( s) > 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in ℤ[ x]. [ABSTRACT FROM AUTHOR]

If L ⊆ ℤ n is a lattice of finite index, let , where the sum is over all lattices L′ with L ⊆ L′ ⊆ ℤ n. Using the automorphic theory of GL2 n, a generalization of the Voronoi-Oppenheim summation formula is proved for these generalized divisor functions. Bessel distributions and Hankel transforms associated with the Shalika model for the degenerate principal series representations of GL2 n(ℝ) are principal tools. [ABSTRACT FROM AUTHOR]

On a smooth compact manifold M, the cohomology of the complex of differential forms is isomorphic to the ordinary cohomology by the classical theorem of de Rham. When M has a Riemannian metric g, the harmonic forms constitute a subcomplex of the de Rham complex because the Laplacian commutes with exterior differentiation. When ( M, g) has no boundary, all of its harmonic forms are closed, and hence the cohomology of this subcomplex is isomorphic to the ordinary cohomology by the classical theorem of Hodge. But when the boundary of ( M, g) is non-empty, it is possible for a p-form to be harmonic without being closed, and some of these, which are exact, although not the exterior derivatives of harmonic p – 1-forms, represent an “echo” of the ordinary p – 1-dimensional cohomology within the p-dimensional harmonic cohomology. [ABSTRACT FROM AUTHOR]

We establish estimates for certain families of character sums over finite fields, including those which arise in Lillian Pierce's recent work [P] on estimating the 3-part of the class number of quadratic fields. [ABSTRACT FROM AUTHOR]