Related papers: Recent progress on the Kakeya conjecture
Much of the science case for the next generation of deep, wide-field optical/infrared surveys has been driven by the further study of dark energy. This is a laudable goal (and the subject of a companion white paper by Zhan et al.). However,…
The aim of this paper is to give an alternative proof of Kac's theorem for weighted projective lines (\cite{W}) over the complex field. The geometric realization of complex Lie algebras arising from derived categories (\cite{XXZ}) is…
This is a survey article on Morse theory based on lectures to graduate students and advanced undergraduates. After a brief review of standard material, mostly without proofs, the Morse theory of complex Grassmannian manifolds is worked out…
This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after…
A Kakeya set $\mathcal{K}$ in an affine plane of order $q$ is the point set covered by a set $\mathcal{L}$ of $q+1$ pairwise non-parallel lines. Large Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya sets with…
I review Chandra's achievements in the last two years in AGN research. I concentrate on some topics of my interest; some others are reviewed in an accompanying article by K. Nandra. I comment briefly on the need to allow for the discovery…
Recent developments in cosmic strings are reviewed, with emphasis on unresolved problems.
We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…
We review some recent developments in mathematical aspects of relativistic fluids. The goal is to provide a quick entry point to some research topics of current interest that is accessible to graduate students and researchers from adjacent…
After surveying some known properties of compact convex sets in the plane, we give a two rigorous proofs of the general feeling that supporting lines can be slide-turned slowly and continuously. Targeting a wide readership, our treatment is…
The purpose of this note is threefold: (i) to recall (with some points made more explicit) the mathematical Weyl algebra model formulation, given before, of the Staruszkiewicz theory of quantum Coulomb field; (ii) to add some new elements…
In this write-up of my SQM 2006 Theory Summary talk I focus on a selection of key contributions which I consider to have a large impact on the current status of the field of strangeness physics or which may have the potential to…
I review at the non-specialist level recent progress in the study of the large-scale structure of the Universe, covering the following areas: (1) Results from recently completed or ongoing redshift surveys of galaxies and X-ray clusters;…
This is a survey article, based on the author's lectures in the 2015 Current developments in Mathematics meeting; published in "Current developments in Mathematics". Version 2, references corrected and added.
I review recent developments in rare and radiative kaon decays from the theory side, with emphasis on those modes that are actively analyzed by the experimental collaborations.
The main purpose of this paper is to make Nakayama's theorem more accessible. We give a proof of Nakayama's theorem based on the negative definiteness of intersection matrices of exceptional curves. In this paper, we treat Nakayama's…
The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and,…
The paper titled "Cremona problem in dimension 2" by W. Bartenwerfer presented a flawed attempt at proving the Jacobian Conjecture. Our aim is to provide a thorough analysis of the author's approach, highlighting the errors that were made…
Some recent developments in the theory of quantum spin systems are reviewed.
In this paper I survey some recent results on finite determination, convergence, and approximation of formal mappings between real submanifolds in complex spaces. A number of conjectures are also given.