Related papers: Notes sur la droite projective de Berkovich

We consider a geometric property of the closest-points projection to a geodesic in Teichm\"uller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This…

These are expanded notes from a four lecture mini-course given by the author at the Spring School on Non-archimedean geometry and Eigenvarieties, held at the University of Heidelberg in March 2023. The course discusses coherent sheaves,…

This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is…

In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the…

Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. For each rational infinite path $c^\infty$ of $E$ we explicitly construct a projective resolution of the…

The projective line over a field carries structure of a groupoid with a certain correspondence between objects and arrows. We discuss to what extent the field can be reconstructed from the groupoid.

This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article…

The purpose of this note is to give the full and self-contained proof of Shchepin's result on a spectral representation of retracts of cubes.

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their…

Let $X$ be a finite-dimensional normed space and let $Y \subseteq X$ be its proper linear subspace. The set of all minimal projections from $X$ to $Y$ is a convex subset of the space all linear operators from $X$ to $X$ and we can consider…

This is the (revised) printed version of the talk no 1056 (june 2012) of the Bourbaki seminar, which will be published in an Ast\'erisque volume. This is a report on a paper by Hrushovski and Loeser (/arxiv:1009.0252). In this paper they…

We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…

This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean…

We give an elementary explicit construction of cell decomposition of the moduli space of projective structures on a two dimensional surface analogous to the decomposition of Penner/Strebel for moduli space of complex structures. The…

The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending…

The purpose of this note is to describe a space that is regular but not completely regular, but only barely so: all closed sets are $G_\delta$-sets and every singleton is a zero-set.

We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.

The classical Besicovitch projection theorem states that if a planar set $E$ with finite length is purely unrectifiable, then almost all orthogonal projections of $E$ have zero length. We prove a quantitative version of this result: if…

Quasi-Polish spaces were introduced by de Brecht as a possibly non-Hausdorff generalization of Polish spaces sharing many of their descriptive set-theoretic properties. We give a self-contained exposition of the basic theory of quasi-Polish…

The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist. List of topics: Euclidean geometry: The Axioms / Half-planes / Congruent triangles / Perpendicular…