历史与综述
This text evolves from the lecture notes for my course on Catalan's conjecture in winter term 2025/26. The ultimate goal is to give full details of Mih\u{a}ilescu's proof. Current chapters: 1. Euler's theorem: $x^2-y^3=1$; 2. V. Lebesgue's…
This communication contributes to research on proof validation as a lens for uncovering didactical phenomena related to proof and proving. We revisit the puzzling case of lower secondary students in France who validate circular proofs. That…
Although inverse functions are introduced early in algebra, many students remain unaware that an inverse expression may legitimately involve a negative root. Instead, they default to assuming a positive root, overlooking the role of domain…
The article focuses on the differences in mathematics performance between girls and boys visible from the first four months of compulsory schooling in the French education system. The influence of gender stereotypes in the evaluation…
This is an exposition of the work of O. Riemenschneider about five ''circles'' of implications relating real analysis theorems each equivalent to the Dedekind completeness of the real field. These circles cover five elements of real…
In this arxiv-post I present my solutions (published or not) to Problems that appeared in Amer. Math. Monthly, Math. Magazine, Elemente der Mathematik and CRUX, that were mostly done in collaboration with Rudolf Rupp. Some of them…
This paper has two goals. The first goal is to show how an extension of second-order logic is a natural framework to formalize portions of Aristotle's \emph{Topics} and to bring to the foreground the logical, linguistic and philosophical…
We discuss and draw the reader's attention to several passages in Vladimir Arnold's note on the epigraph to the novel in verse "Evgenii Onegin" by A$.$S$.$Pushkin, as well as to puzzles hidden in the novel by the poet himself.
This article describes Boy's surface in a nice way that does not make many demands on three-dimensional visualization. The article includes a kit that you can print out onto card stock and assemble with scissors and tape.
We introduce \emph{patterned numbers}, a digit--divisor-based classification of integers motivated by recreational mathematics. A number is defined to be patterned if at least one of its positive divisors appears as a digit in its base-10…
In the 1960s, John Nash proposed a method to resolve singularities. Five decades of encouraging results could not prevent an unexpected ending: the method does not work in general. In this note (written in Spanish), we tell the story of the…
This article aims to provide a comprehensive overview of sparse optimization, with a focus on both sparse signal recovery and sparse regularization techniques. We will begin by exploring the foundations of sparse optimization, delving into…
This article describes our invention of a new poetic form based on projective geometry. In doing this we also explore the 'what ifs' in mathematics and poetry which spark the creative processes of poet and mathematician. In other words,…
This is a slightly edited translation of a paper in Dutch which appeared in Nieuw Archief voor Wiskunde (5) 25 (2024), No.2, 87-90 on the occasion of I.G. Macdonald's death in 2023, and aimed at a very broad mathematical audience. First we…
This paper describes how the discrete Askey scheme independently arose in Russia and how Askey learned about this. In particular, Askey met main characters in this story, namely Gel'fand and Suslov as well as Nikiforov and Uvarov, during…
We present the Legendre transformation in a geometric way based on the procedure of the Legendrian lift. This approach allows us to understand some interesting properties of it, in particular, the reason for the appearance of singularities…
In this expository paper we present some ideas of algebraic topology (more precisely, of homology theory) in a language accessible to non-specialists in the area. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is…
In this expository note we show how combinatorial Nullstellensatz by N. Alon naturally appears in solutions of elementary problems. Simple ideas gradually and naturally appear in such solutions, thus bringing a reader to generalizations.…
We present a short elementary proof of the well-known criterion for a cubic polynomial to have three real roots. The proof is based on Fermat's approach to calculus for polynomials. This approach illustrates the idea of a derivative…
We present short elementary proofs of the well-known Ruffini-Abel-Galois theorems on insolvability of algebraic equations in radicals. These proofs are obtained from existing expositions by stripping away material not required for the…