相关论文: A streamlined proof of the Kakeya set conjecture i…

We give a detailed outline of the proof that the Kakeya conjecture follows from the sticky case. This proof is due to Wang and Zahl and appears in a recent paper. The sticky case was proven in earlier work of Wang-Zahl, building on an…

If S is a smooth compact surface in $\mathbb{R}^{3}$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3$, $\left\|E_S f\right\|_{L^p\left(\mathbb{R}^3\right)}…

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…

We prove the Kakeya set conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. This entails extending and combining the techniques of Arsovski [Ars21a] for $N=p^k$ and the author and Dvir [DD21] for…

We prove a refined trilinear Kakeya estimate in three dimensions, valid for small values of the transversality parameter.

We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery, and Tao.

We prove Union-Closed sets conjecture.

We prove Simon's conjecture for 3-manifolds.

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.

We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.

We present a simple and purely combinatorial proof of Steel's result according to which the core model is \Sigma^1_3 correct under the appropriate hypotheses.

We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{{3 \over 2}})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines…

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…

In this note, we present a simple directed graph proof of Sharkovsky's theorem.

We give a new simpler proof of a theorem of Jayne and Rogers.

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least…

In this note we verify the equivalent version of Huppert's conjecture for $K_3$-groups.

We provide a short proof of the 1-dimensional flat chain conjecture.

We prove an elementary lemma concerning primitive amalgams and use it to greatly simplify the proof of the Sims conjecture in the case of almost simple groups.

We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of…