4,287 papers in this slice of arXiv.
Benjamin Collas
Large language models have begun refuting long-standing conjectures and, for a few thousand dollars of tokens, solving long-open problems (OpenAI, August 2026). The introspection this has prompted about the future of mathematical discovery is overdue, and the anxiety accompanying it legitimate -- but both are attached to the wrong loss. What machines now produce is the countable part of mathematics -- theorems, proofs, refutations -- which was always the residue of the work, not its product. The product is human understanding: not a stock of results but a collective, hard-won way of deciphering the world and acting upon it. The two are arcs of a single loop -- looking produces the residue; taking it up again, one journey at a time, is what rebuilds the shared understanding. Machines are strong on the countable arc, absent from the one that feeds it. The peril is to leave the loop open. AI did not create the confusion between residue and product; it has called a bluff long on the books, driving the cost of the residue towards zero and making the scarce thing visible at last. The pressing questions are therefore institutional: who can check the claims of AI companies, what the work becomes for the next generation of researchers, and whether the one thing that cannot be mass-produced -- the journey that nourish a shared understanding -- continues to be funded. Mathematics, we argue, is uniquely placed among the sciences on the first question -- a proof answers to no one's permission -- and uniquely exposed on the last: the journey has never had a price our institution knew how to pay. The decision is ours.
Andrés Navas
The panmagic square engraved in a temple in Khajuraho, India, and studied by Narayana Pandita in the 14th century, closely guards a secret: its group of symmetries is isomorphic to that of the hypercube, the four-dimensional analogue of the standard cube.
Matthew Baker
The year 2026 marks the 150th anniversary of a remarkable achievement in mathematics: in 1876, the French mathematician Édouard Lucas showed that the 39-digit number M127:=2127−1 is prime. This stood for 75 years as the largest known prime, and it remains the largest prime number discovered and certified without the aid of a mechanical device. We review the history behind this discovery and give a modern proof of the closely related Lucas-Lehmer test, which is still the main engine behind the certification of large prime numbers.
Scott Bollt, Mario di Bernardo, Jeremie Fish +4
In this paper, we give a memorial tribute to Prof. Erik M. Bollt (1967--2025).
Max Weinreich
In this essay, I present the case for total opposition to the use of artificial intelligence in mathematics. I offer proposals for how individuals, departments, journals, and institutions can act in concert to make sure that mathematics survives the coming crisis.
Chandradew Sharma
Why does (-)x(-)=+? Although every student learns this rule, many accept it as a convention rather than as a mathematical necessity. This article develops a conceptual explanation by tracing the evolution of number from counting objects to representing opposites. Beginning with the emergence of numerals, place value, zero, and negative numbers, it shows how each stage extended the meaning of number and the operations defined on it. The discussion argues that multiplication of signed numbers is most naturally understood in terms of preserving or reversing direction, so that the familiar rule (-)x(-)=+ follows as a logical consequence of a coherent arithmetic rather than as an arbitrary convention. By combining historical development with mathematical reasoning, the article offers an accessible framework for teachers and students seeking a deeper understanding of signed multiplication and illustrates how the evolution of mathematical ideas can enrich classroom learning.
Oliver Knill
This is a snapshot of a first part on a possibly much longer text on finite geometries, meaning graphs or finite abstract simplicial complexes. In in this first batch we review 12 subjects: Gauss-Bonnet, Poincare-Hopf, Index expectation, Euler's gem, Euler-Poincare,Unimodularity, Brouwer-Lefschetz, Sphere formula, Level sets, Index formula, Quadratic cohomology and Higher characteristic.
Borbála Neogrády-Kiss
This paper presents an autonomous online application designed to support inquiry-based learning in mathematics when in-person instruction is not possible. The application teaches the concept of boundedness of functions through adaptive questioning, requiring active problem solving without teacher intervention. It was tested with 92 Hungarian secondary school students preparing for the advanced-level final examination. Assessment based on Bloom's taxonomy showed that most students achieved at least a medium level of understanding, with many reaching higher cognitive levels. The results suggest that such applications may support learning and complement traditional instruction. Moreover, the study suggests that future AI-based educational systems may benefit from similar inquiry-oriented approaches, emphasizing active thinking and guided problem solving rather than passive answer generation.
Vladimir A. Vladimirov
The purpose of the present paper is to advertise and illuminate the applied research of Mikhail Alekseyevich Lavrentiev, a distinguished scientist who generated new ideas and obtained many outstanding results in mechanics and mathematics. He also initiated and maintained numerous flourishing activities in mechanics and applied mathematics in the former Soviet Union. He was notably an outstanding researcher in pure and applied mathematics as well as in engineering. On the applied side, his research spanned fluid dynamics, explosion physics, mechanics, technology, and several branches of engineering. It was always aimed at the practical challenges of his time and achieved impressive successes. It has been fascinating to uncover all of this. Much of his biographical material and a review of his activities in pure mathematics have been published, but no review of his applied achievements is available. This paper aims to give Lavrentiev's applied research the recognition it deserves in the communities of fluid dynamics, applied mathematics, engineering, and the history of science.
James M. Hyman
Large systems of linear equations often contain far fewer active degrees of freedom than their ambient dimension suggests. A particularly transparent instance occurs when a large matrix factors as A = TSW, where T is m x n, S is n x n, and W is n x m, and m >> n. The action of A then passes through an n-dimensional intermediate space. Sylvester's theorem relates the nonzero eigenvalues of a product XY to those of the reversed product YX. Applied cyclically, it shows that the nonzero spectrum of the large matrix A is completely determined by either of the small matrices B = (WT)S, C = S(WT). This article develops the theorem from first principles and explains the geometry behind the factorization. The result yields exact reduced systems, both for linear algebraic equations and for first-order linear ordinary differential equations. Worked examples carry this through in detail: spectral reduction, reconstruction of full-space eigenvectors, reduced solution of shifted systems, and exact integration of a three-dimensional ODE through a two-dimensional model. A separate section takes up singular values. Those of A, B, and C do not agree in general, though they do when the bases are orthonormal, a case the same theorem settles once it is applied to A*A. The closing sections separate exact factorization from projection-based approximation, and flag what the theorem leaves undetermined: zero-eigenvalue structure, conditioning, and transient behavior. The material is classical. What this article adds is one continuous development, from first principles through to the limits of the theorem, pitched for an upper-level undergraduate course.
Nikhil Bansal, Aleksandar Nikolov
Combinatorial discrepancy theory is a subject with roots in combinatorics, geometry, and number theory, and with numerous applications to mathematics and computer science. At its core, discrepancy theory is about dividing a collection of objects into two parts that are as balanced as possible. For example, given a collection of subsets of a finite universe, we may wish to color the elements with two colors so that each set is approximately evenly split. Other problems in discrepancy are more geometric in flavor, and ask for example, to assign signs to a collection of vectors, so that the sum of the signed vectors is as small as possible. Classical results, such as the Beck-Fiala theorem and Spencer's "six deviations" result, show that it is often possible to attain remarkably small discrepancy, often far smaller than what naive random colorings achieve. In recent years, discrepancy theory has undergone a transformation, driven by new algorithmic techniques and a rich interplay between probability, optimization, and convex geometry. These developments have led not only to new constructive proofs of foundational theorems, but also to several new results and research directions. This monograph aims to provide an accessible and unified introduction to these modern developments, with a focus on the core algorithmic and convex geometric ideas that have driven them. For several results, we provide new simpler analyses, while highlighting the intuition behind the proofs.
Amir Moradifam
Mathematics is often valued for its role as the language of science, but it is equally important as a human practice that teaches us how to think and cultivates creativity, problem-solving skills, attention, and judgment. The rapid rise of modern AI systems, which can solve many standard problems and produce fluent mathematical explanations, forces a reconsideration of the nature of mathematical understanding and how it is developed. This article reflects on the impact of AI across three intertwined domains: mathematics education, mathematical research, and the broader role of mathematics as a way of learning to think. In education, AI changes how students interact with mathematical material, making high-quality assistance widely available while weakening the traditional link between assigned work and demonstrated understanding. The challenge concerns not only academic integrity but also the risks of cognitive offloading and the illusion of explanatory depth. In research, AI can accelerate exploration in various ways, including generating conjectures and examples, performing symbolic manipulation, suggesting proofs, and assisting in the communication of ideas. It has also begun to contribute to discoveries on problems that had resisted human efforts. Yet its limitations remain serious, especially in long arguments requiring global structure, careful verification, and the synthesis of multiple ideas. The central claim is that in an era when routine tasks become easier to automate, the most human aspects of mathematics, including conceptual insight, problem framing, genuine understanding, and creative judgment, become more visible and more valuable.
Jozef H. Przytycki
This paper is an extended version of two talks I gave during workshop ``Loops'13" in Bedlewo in June 2023. In the first talk I gave a historical introduction to Knot Theory. In the second, I traced my journey toward Yang-Baxter homology and this talk has a partially survey and a partially novel character.
Aravind Asok, Tom Bachmann, Michael J. Hopkins
Given a pair of complex algebraic varieties, we can ask: does every analytic map admit an algebraic representative? Such questions are most interesting when the source and target are non-compact varieties, in which case the problems are closely linked with Hodge-type conjectures. Concrete avatars of this kind of problem, which we discuss from several points of view, include: which holomorphic vector bundles on an affine variety admit an algebraic structure, and which homotopy classes of maps admit algebraic representatives?
Petra Schwer
Reading a math textbook isn't like reading a novel. You've probably already figured that out on your own. But how *do* you read mathematical texts? And how can you actually learn from a book or lecture notes? This document is a guide to independently reading and working through mathematical texts. Originally written as supplementary material for a flipped classroom course for first year math students it may serve as a detailed reading guide for anybody interested in math.
Clifton Callender
Infinite Canons is an ongoing series of canons with infinite solutions. More specifically, each canon is based on a melodic line that can be combined in any number of voices, at any tempo ratios (rational or irrational), and with each voice moving either forward or in retrograde inversion, while maintaining harmonic consistency. This paper describes the structure of these maximally self-similar melodic lines based on two different constructions: 1) a discrete prime-factorization approach yielding self-similarity under all rational ratios, and 2) a continuous logarithmic approach extending this to irrational ratios. In both cases the vertical interval between voices in a tempo ratio of λi/λj is given by a homomorphism φ(λi/λj), which is a constant independent of time. Furthermore, under these constructions retrograde inversion collapses to transposition, allowing for all manner of table canons. These structures are demonstrated with suggestive realizations of several different infinite canons. Future work includes a more complete mathematical treatment, musical applications, and an interactive program that allows users to explore an unlimited number of realizations of these pieces.
Alexis Langlois-Rémillard, Mia N. Müßig, Érika Roldán
We present results around an isoperimetric problem built on polyforms: What is the biggest enclosed area one can build using polyforms in each of the three plane tessellations? We give challenges to the readers and present Shimauchi's proof of the biggest area a fence made of pentominoes can enclose. A translation of the instance using integer linear programming is also given. A companion web app is available to test some of the challenges we propose and for activities, and we included extra pages with a cutout handout of the board and pieces of the puzzles, so that you can print and cut them to read this paper hands on.
Math Dicker
In Proposition II of his manuscript, Galois writes the well-known remark: Il y a quelque chose à compléter dans cette démonstration. Je n'ai pas le temps. Although Galois did not complete the proof, it is possible to reconstruct the essential content of Proposition II and to supply the missing arguments. As usual, V denotes the linear form ax1+bx2+cx3+..., where x1,x2,x3,... are distinct roots of a separable polynomial. Let L be the corresponding splitting field over a ground field of characteristic zero. On the one hand, Proposition II concerns the factorization of the minimal polynomial g(x) of V over an intermediate field M contained in L; on the other hand, it concerns the factorization of the same polynomial into irreducible factors whose coefficients belong to the intermediate fields conjugate to M. It is precisely this latter aspect that constitutes the central theme of Proposition II. The notions of 'groupe de permutations' and 'groupe de substitutions' are of fundamental importance in the Mémoire. We provide a characterization of these notions in modern terminology. Associated with every 'groupe de permutations' is a polynomial whose coefficients are invariant under the corresponding 'groupe de substitutions'. Moreover, a 'groupe de permutations' determines a partition of the Galois group, making it possible to factor the minimal polynomial g(x) into factors whose coefficients belong to the intermediate fields corresponding to the associated 'groupes de substitutions'. Finally, we prove that the substitutions of the Galois group are field automorphisms of the splitting field L. This establishes the connection between Galois' original formulation and the modern formulation of Galois theory.
Sylvie Monniaux
The content of the following pages was part of a special topics lecture given at the Australian National University during the first semester of 2026. That was a very nice expericence, I really enjoyed giving this 12 weeks (2 hours a week) lecture. It is meant to be self contained, starting with results on Fourier transform and Sobolev spaces. As a toy model, before treating the Navier-Stokes system, we focus on the non linear heat equation where the non linearity is polynomial. Ultimately, we prove existence and uniqueness of mild solutions of the Navier-Stokes equations in critical spaces. This script contains some exercises and the text of the mid-semester exam as well as the final exam.
Oleg Zubelevich
This expository article serves as a methodical guide, presenting a rigorous mathematical formulation of the D'Alembert-Lagrange principle and the derivation of the Lagrange equations for mechanical systems subject to ideal constraints. Designed for educational purposes, the paper establishes a clear geometric framework for the extended phase space, defining the space of virtual displacements and constraint reactions independently of their specific analytical representations. A central focus of this instructional text is the transition from the general equation of dynamics to the Lagrange equations of the second kind for holonomic systems. Specifically, we demonstrate that the Lagrange equations can be naturally and elegantly derived from considerations of the covariance of the variational derivative under the embedding mapping of the configuration manifold.