相关论文: Recent progress on the Kakeya conjecture
We survey progress on the Kakeya conjecture in Euclidean space, with an emphasis on developments that have occurred since the previous surveys by Wolff and Katz-Tao.
Roughly speaking, the Kakeya Conjecture asks to what extent lines which point in different directions can be packed together in a small space. In $\R^2$, the problem is relatively straightforward and was settled in the 1970s. In $\R^3$ it…
The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and…
Recently, Hong Wang and Joshua Zahl announced a proof of the 3-dimensional Kakeya conjecture. This is a survey article on the proof of Kakeya. We introduce the problem, discuss previous work and some of the difficulties of the problem, and…
The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these…
We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's strengthened formulation of the maximal…
This is a Seminaire Bourbaki survey of the proof of the Kakeya conjecture in three dimensions. The survey is written for a broad mathematical audience. We sketch all the ideas in the proof, with many pictures.
In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the $N^{3/2}$-barrier, proving that a set of $N$ well-spaced…
We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we…
Recent years have seen dramatic progress in cosmology and particle astrophysics. So much so that anyone who dares to offer an overview would certainly risk him- or herself for being incomplete and biased at best, and even incorrect due to…
A Kakeya set is a compact subset of $\mathbb{R}^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff and Minkowski dimension $n$. There is a special class of…
We survey recent developments on the Restriction conjecture.
This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin neighborhoods of line segments that point in different…
This is a survey on Kawaguchi-Silverman conjecture.
In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the Andr{\'e}-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, stable modular forms,…
We prove an x-ray estimate in general dimension which is stronger than the Kakeya estimates of Wolff. This generalizes an x-ray estimate in three dimensions which is also due to Wolff.
In this paper, we give a survey of the recent develpoments of the DDVV conjecture.
We survey Kondrat'ev--Landis' conjecture, providing an up-to-date account of the main advances and describing the techniques developed. We complement the overview with references and formulations of the problem in further closely connected…
This paper is an attempt to survey the current state of our knowledge on the Caccetta-Haggkvist conjecture and related questions. In January 2006 there was a workshop hosted by the American Institute of Mathematics in Palo Alto, on the…
The goal of this paper is to review the advances that were made during the last few decades in the study of the entropy, and in particular the entropy method, for Kac's many particle system.