Related papers: The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb…

In this note, we consider Szemer\'{e}di's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a…

This work consists of two parts. In the first part we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces.…

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.

Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof…

Let $X$ be a connected, noetherian scheme and $\mathcal{A}$ be a sheaf of Azumaya algebras on $X$ which is a locally free $\mathcal{O}_{X}$-module of rank $a$. We show that the kernel and cokernel of $K_{i}(X) \to K_{i}(\mathcal{A}) $ are…

Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded…

We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.

Additive combinatorics asks for lower bounds on sumsets and restricted sumsets over finite fields. Central examples are the Cauchy-Davenport theorem and the Erd\H{o}s-Heilbronn conjecture. In this note, we develop Das's linear algebraic…

We prove a generalization of Tukia's ('85) isomorphism theorem which states that isomorphisms between geometrically finite groups extend equivariantly to the boundary. Tukia worked in the setting of real hyperbolic spaces of finite…

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian $\beta$ Ensembles and the large $N$ expansion of the densities" with the use of the topological recursion…

Recently, the first two authors proved the Alon-Jaeger-Tarsi conjecture on non-vanishing linear maps, for large primes. We extend their ideas to address several other related conjectures. We prove the weak Additive Basis conjecture proposed…

The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemer\'edi [E-S]. (see also…

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach…

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher dimensional theories of gravity. We extend the results shown by R. Bartnik and G. Fodor for the 3-dimensional case in…

We propose to study the restriction conjecture using decoupling theorems and two-ends Furstenberg inequalities. Specifically, we pose a two-ends Furstenberg conjecture, which implies the restriction conjecture. As evidence, we prove this…

Menger's basis property is a generalization of $\sigma$-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non $\sigma$-compact sets of reals with Menger's property.…

Zagier-Hoffman's conjectures predict the dimension and a basis for the $\mathbb Q$-vector spaces spanned by $N$th cyclotomic multiple zeta values (MZV's) of fixed weight where $N$ is a natural number. For $N=1$ (MZV's case), half of these…

In 2021, the first author and Kalita obtained two general hypergeometric formulas for sums involving certain rising factorials to prove some supercongruence conjectures of Guo related to (B.2) and (C.2). In this paper, we further generalize…

We prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the…

In ["Illumination of convex bodies with many symmetries", Mathematika 63 (2017)], Tikhomirov verified the Hadwiger-Boltyanski Illumination Conjecture for the class of 1-symmetric convex bodies of sufficiently large dimension. We propose an…