Related papers: On a positive equicharacteristic variant of the $p…

We present several versions of the Jacobian Conjecture in positive characteristic each of which if true would imply the Jacobian conjecture in characteristic 0. We test these characteristic p versions of the conjecture against several…

The $p$-curvature of a system of linear differential equations in positive characteristic $p$ is a matrix that measures how far the system is from having a basis of polynomial solutions. We show that the similarity class of the…

In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group.Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic $p$-curvature…

We give a counterexample to a recently conjectured variant of the Penrose inequality.

The aim of this article is to give a new proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case.

In this short note, we prove the equivalence of Grothendieck-Katz $p$-curvature Conjecture with Conjecture F in Ekedahl-Shepherd-Barron-Taylor. More precisely, we show that Conjecture F implies the $p$-curvature conjecture, and that the…

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.

We introduce p-equivalence by asymptotic probabilities, which is a weak almost-equivalence based on zero-one laws in finite model theory. In this paper, we consider the computational complexities of p-equivalence problems for regular…

We introduce a variant of the $C_p$ condition (denoted by $SC_p$), and show that it characterizes weighted weak type versions of the classical Coifman-Fefferman and Fefferman-Stein inequalities.

The aim of this paper is to gain a better understanding of weak and strong positivity for exterior forms on complex vector spaces. We prove a dimensionality reduction argument for positive forms, which allows us to restrict to the case of…

We study probabilistically informative (weak) versions of transitivity, by using suitable definitions of defaults and negated defaults, in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults…

The Grothendieck--Katz $p$-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo $p$ has vanishing $p$-curvatures for {\em almost all} $p,$ has finite monodromy. It is known that it suffices to prove…

The aim of this paper is to bring together a new type of quantum calculus, namely $p $-calculus, and variational calculus. We develop $p $-variational calculus and obtain a necessary optimality condition of Euler-Lagrange type and a…

Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\geq0$ and $0<p\leq1$, then \[\Vert N\Vert_{p,\infty}\leq2\Vert…

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every…

In this paper we study Hardy and Poincar\'e inequalities and their weak versions for quadratic forms satisfying the first Beurling-Deny criterion. We employ these inequalities to establish a criticality theory for such forms.

We state and prove a variant of the Andr\'e-Oort conjecture for the product of 2 modular curves in positive characteristic, assuming GRH for quadratic fields.

We establish the equivalence between weak and viscosity solutions for non-homogeneous $p(x)$-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution…

Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider Alperin's weight conjecture (over $k$) from the point of view of (stable)…

We prove a sharp quantitative form of isocapacitary inequality in the case of a general $p$. This work is a generalization of the author's paper with Guido De Philippis and Michele Marini, where we treated the case of $2$-capacity.