Related papers: Ergodic bounded constructions, Disjointness and We…

We prove that, if $B_1, \dots, B_n$ are disjoint bases of a rank-$n$ matroid, then there are at least $\lfloor{\frac{n}{6 \lceil{\log n}\rceil}}\rfloor$ disjoint transversals of $(B_1, \dots, B_n)$ that are also bases.

We study the generalizations of Jonathan King's rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z^n-actions. We prove that these results remain valid in the case of…

We study a one dimensional Lagrangian problem including the variational reformulation, derived in a recent work of Ambrosio-Baradat-Brenier, of the discrete Monge-Amp\`ere gravitational model, which describes the motion of interacting…

We study a class of maps having the Collatz function (famously related to the Collatz Conjecture) as an example, under the topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key…

In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a…

Let $X$ be a complex manifold containing a hypersurface $D$ and let $D^s$ denote the singular locus. We study the problem of extending a flat connection with logarithmic poles along $D$ from the complement $X \setminus D^s$ to all of $X$.…

We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was…

This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated…

I give a simplified proof of one of the main results in the recent preprint arXiv:1311.5858 by X. Luo and K. Zuo. Furthermore, I discuss the relation with the Coleman-Oort conjecture on special curves in the Torelli locus.

In this note we present a proof of multiple recurrence for ergodic systems (and thereby of Szemer\'edi's theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs-de Leeuw-Glicksberg decomposition…

We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu…

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally…

We study the Carnot theorem and the configuration of points and lines in connection with it. It is proven that certain significant points in the configuration lie on the same lines and same conics. The proof of an equivalent statement…

We prove that the group of birational automorphisms of a geometrically irreducible algebraic surface over a finite field is Jordan. We show that the analogous statement fails in higher dimensions. Finally, we prove that groups of birational…

For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is…

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.

Over Cohen--Macaulay rings admitting a pointwise dualizing module, we show that the class of modules of restricted projective dimension bounded by any integer is finitely deconstructible and that the class of modules of restricted flat…

In this paper the stability and the perturbation bounds of Markov operators acting on abstract state spaces are investigated. Here, an abstract state space is an ordered Banach space where the norm has an additivity property on the cone of…

A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence…

Mixed (asymmetric) orthogonal arrays (MOAs) generalize classical orthogonal arrays by allowing columns over different alphabets. However, their study requires very different structural tools than those used for symmetric orthogonal arrays…