Mathematics

We classify all left-invariant real affine connections in dimension three. Our approach reduces the three-dimensional problem to a two-dimensional one by decomposing each left-invariant affine connection into a two-dimensional part and an…

We consider the Tanaka-Webster geometry of surfaces embedded in a 3-dimensional Lie group with a CR structure inherited by a contact form. We define the notions of Gauss and mean curvature and give specific examples.

Let $K/\mathbb Q_p$ be finite and let $f\in\mathcal O_K[X]$ be monic, of degree at least two, with $f'(X)\in\mathfrak m_K\mathcal O_K[X]$, equivalently $\bar f\in k[X^p]$. For a compatible inverse branch $f(t_{n+1})=t_n$ with…

We introduce a binary matroid approach to the enumeration of mod 2 toric-colorable seeds of fixed Picard number. We organize these matroids by their contraction category and enumerate weak pseudomanifold subcomplexes by a dynamic…

It is well known that solutions of elliptic equations inherit analyticity from analytic coefficients, while much less is understood about the inheritance of ultra-analytic regularity, especially for nonlocal equations. This paper develops a…

Given $n\in \mathbb{N}$, $p\in [1,\infty)$, and a weight $\gamma$ satisfying the local Muckenhoupt $A_p$ condition, we introduce a weakened version of the Ahlfors--David codimension-$\theta$ regularity condition for Ahlfors--David…

Let $A$ be a finite non-empty subset of an abelian group $G$, and let $r_A(d)=|\{(a,a')\in A^2:a-a'=d\}|$. Croot and Lev asked whether the pointwise half-threshold condition $r_A(d)\ge |A|/2$ for every $d\in A-A$ forces $A-A$ to be either a…

We prove a Schur--Zassenhaus theorem for finite skew braces. More precisely, if \(B\) is a finite skew brace and \(I\) is an ideal of \(B\) such that \(|I|\) and \(|B/I|\) are coprime, then \(I\) admits a complement in \(B\).

A brief overview of results concerning the connection between the Hilbert-Polya conjecture and the Riemann hypothesis about the Riemann zeta function, some new results on p-adic quantum computing, quantum entanglement based on lattice spin…

We prove fixed finite-chaos-order estimates for Hilbert-space-valued Gaussian random tensors. Given a finite-rank kernel \[ K\in\cA_1\otimes\cdots\otimes\cA_m\otimes\cC\otimes\cE \] and the associated decoupled homogeneous Gaussian chaos…

Parameter estimation in nonlinear dynamical systems from observational data is a fundamental inverse problem with applications in many disciplines. In practice, this is further complicated by the fact that observations are often noisy,…

In this paper, we prove an approximation and interpolation theorem for maxfaces in the Lorentz--Minkowski $3$-space $\mathbb{L}^3$. Alarc\'on, Forstneri\v{c}, and L\'opez established approximation and interpolation theorems for conformal…

In this note, we give a Tur\'an theorem for Cayley graphs $\Cay(\Z_p,S)$ over prime cyclic groups $\Z_p$. For a graph $F$ and a finite abelian group $G$, define the Cayley--Tur\'an number by \[ \exCay(F,G) = \max\{|S|:S=-S\subseteq…

Let $p$ be an odd prime number and let $\overline{\mathbb{F}}_p$ be a fixed algebraic closure of the finite field of order $p$. Let $K$ be a global function field of characteristic different from $p$ and let $G_{K}$ be the absolute Galois…

The purpose of this paper is to give a natural divisor-theoretic formulation of the counting method introduced by Halburd for computing degree growth, in a form applicable to birational dynamical systems on varieties of arbitrary dimension.…

Let $M$ be a closed orientable 3-manifold and $k$ be a knot in $M$. Then the Dehn surgery of $M$ along $k$ with slope $\alpha$ is not surface fibered for all but a sparse set of slopes.

In the category of digroups there is a natural pretorsion theory in which the torsion-free digroups are all groups, and torsion digroups form a category isomorphic to the category of non-empty sets. It is also possible to extend the theory…

Let $D$ be the ring of $S$-integers in a global field and $\da$ its profinite completion. We propose a profinite version of the Bateman--Horn conjecture over $D$ and provide a first comparison with the classical one and its generalizations.…

The celebrated Chern conjecture asserts that any closed minimal hypersurface in $\mathbb{S}^{n+1}$ with constant scalar curvature is isoparametric. In this paper, we resolve this conjecture in the affirmative for $M^4 \subset \mathbb S^5$…

An $(n,R)$-covering sequence over a finite alphabet $\Sigma_q = \{0,1,\dots, q-1\}$ is a cyclic sequence whose consecutive length-$n$ windows form a covering code of radius $R$. Equivalently, every word in $\Sigma_q^n$ is within Hamming…