Papers

The aim of this note is to introduce natural geometry, in which the rectilinear plane is a primitive. This plane is admitted together with its rectitude, and its rectitude states that a Lambert quadrilateral has its fourth angle right. The fourth angle of a Lambert quadrilateral may be right, acute or obtuse, and this trichotomy distinguishes three planes. Starting from the rectitude, we prove the uniqueness of the parallel by a chain of neutral steps: the rectangle, the triangle whose angles sum to $\pi$, and Legendre's theorem \cite{legendre}, all borrowed from the classical literature. That uniqueness is equivalent to the fifth postulate, which we thus reach in Euclid's own wording: the equivalence is read in the natural direction, and it is not a derivation of the fifth postulate from the other four, since the independence of that postulate is not in question. In the rectilinear plane the parallels are characterised as exactly the images of a line under a point-reflection, and equivalently as the pairs of equidistant lines. The other two planes are then placed beside the rectilinear one, and the same definitions serve for all three.

As its 40th anniversary nears, we present a history of the International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), through interviews with figures from its past, present, and future.

We propose that mathematics is done largely as a means of both coping with and celebrating our physicality in the natural world. Inspired by David Abram's "Becoming Animal", we suggest that – in opposition to the popular image of an otherworldy Platonic realm– the spiritual component of doing mathematics is grounded on and in the Earth. We explore this as a means of questioning the widespread adoption of AI in the workflows of research mathematics.

This paper presents a conceptual, cognitive-constructivist model for teaching the half-derivative. The model starts from a student's question and frames it as the problem of generalizing the familiar operation of differentiation. On a suitably chosen space of monomials, under the assumptions of linearity and of lowering the exponent by one half, the requirement that the operator applied twice give the ordinary derivative produces a recurrence for the coefficients. The freedom of normalization becomes visible, and an additional convention yields a formula compatible with the monomial rule of the Riemann-Liouville half-derivative. The domain of the operator and its relation to the Riemann-Liouville and Caputo operators are treated separately. The mathematical results rest on the author's earlier preprints; the contribution of this paper is their transformation into an instructional pathway organized in phases, with diagnostic questions, an analysis of typical errors, and comparative problems. The model is a design proposal: its effectiveness has not yet been established by systematic empirical research. The interactive Socratic implementation and the detailed diagnostics are the subject of Part II.

The Sachdev–Ye–Kitaev (SYK) model is a quantum many-body model of Majorana fermions with all-to-all random interactions. It has attracted wide interest in condensed matter and high-energy physics as a solvable model of non-Fermi liquids and holography. Fix an even integer $q\geq4$, and let $Z_N(\beta)$ be the partition function of the Gaussian $q$-body SYK model. We prove that for all sufficiently small inverse temperatures $\beta > 0$, there exists $\rho=\rho(\beta)>0$ such that \[ N^{q/2 -1}(\log Z_N(\beta)-\mathbb{E}[\log Z_N(\beta)]) \Longrightarrow \mathcal{N}(0,\rho^2). \] In addition to the mathematical results, we discuss how the proofs were developed and refined through interaction between human researchers and AI, and we describe the subsequent formalization of the main arguments in Lean 4. This work thus serves as a case study of a research workflow that integrates human insight, AI-assisted exploration, iterative proof refinement, and formal verification.

We present short proofs for the main parts of the recently completed solution of Erdős Problem 506, which asks for the minimum number of distinct circles determined by a finite planar point set that is neither collinear nor concyclic. An inversion argument and a seven-point line lemma show that eight points determine at least $17$ circles, and an explicit configuration attains this bound. A single argument combining Melchior's inequality with Langer's incidence bound establishes the sharp formula for every $n\ge15$. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.

On 6 October 2026 OpenAI published an initial 722 AI-written mathematical manuscripts, organized into 372 problem families, in the public repository github.com/openai/math. The release includes claimed solutions to many prominent open problems. We approach this collection as a treasure mine of mathematical ideas. Our aim is to extract new lemmas, constructions, inequalities and proof mechanisms introduced in the source arguments, and to distinguish them from established tools and new combinations of those tools. We call these mathematical instruments. Extraction comes first; their validity, novelty and possible reuse then become subjects for further study. The resulting descriptions also help readers understand the selected papers without having to reconstruct every argument at the outset. We examine 38 families in two groups of 17 and 21, spanning number theory, geometry, analysis, combinatorics, complexity, algebra, operator algebras, topology and mathematical physics. Each entry records the claim, extracted instruments, antecedents and caveats. Eleven cross-cutting patterns connect the entries, and a scoreboard records formalization status and dependencies. Novelty assessments are provisional and sometimes mixed. Selected source comparisons correct mathematical statements, hypotheses and counts; they do not independently establish priority or certify the underlying proofs. No Lean development was compiled or kernel-checked for this survey. We also briefly describe a subsequent application of the signed restriction lemma from family 112 to circuit lower bounds at every fixed depth.

We demonstrate the well-known fact that the diagonal of a unit regular pentagon has length the golden ratio. While similar to other proofs, we provide a novel figure that we believe is particularly well-suited to achieve full understanding at a glance.

Fractions require learners to revise properties associated with whole numbers. This paper presents a design rationale and a critical analysis of an eight-level source progression for fractions (Grades 1-8, nominal ages 6-14), from which the French-curriculum and International Baccalaureate chapters published at https://commeunjeu.org are assembled. Its central hypothesis is a bridge from sharing to numerical measure: a share of one unit is represented by a bar, read as a length relative to a specified unit, and located on the number line. The reference unit is kept stable in the initial tasks to focus attention on fraction magnitude, equivalence and operations; changing the unit is deliberately postponed, without treating this as a mathematical requirement or an established reduction in cognitive load. Three principles organise the meanings: sharing one unit, coordinating bars and number lines, and establishing fractions as numbers before explicitly treating them as scalar operators. Two supporting principles concern valid local justifications before formal generalisation and chains of equalities, introduced in Grades 2-3, with written reasons in lessons from Grade 4. The analysis relates these choices to the research literature and selected curricula, examines task strategies and feedback, and distinguishes realised features from discrepancies and proposed revisions. The source documents and their curriculum projections do not establish the order actually experienced by pupils. No empirical evaluation is reported. Primary hypotheses address transfer between representations, coordination of units and the numerical status of fractions before operators; the audit also identifies limits to the reproducibility of its findings.

We consider the future of mathematics as a profession in a world where artificial intelligence may surpass human mathematicians in essentially every technical task. Through an informal dialogue, we explore why human mathematical expertise might nevertheless remain valuable, what kinds of expertise universities should cultivate, and how graduate education might adapt. We also discuss the institutional and economic uncertainties that threaten the continued existence of a thriving mathematical profession, even if the value of human mathematical understanding endures.

These are notes to accompany my public lecture at the International Congress of Mathematicians, in Philadelphia in July, 2026. The goal is to discuss some developments at the interface of AI and mathematics in the context of the long conversation of mathematics.

Philip L. Roe, the renowned British scientist, passed away on April 26, 2026, at the age of 87, while Professor Emeritus at the University of Michigan. My association with Phil Roe began in 1984, when he joined the Cranfield Institute of Technology, UK, as Professor of Computational Fluid Dynamics, while I was a postdoctoral research fellow at the same institution. For more than four decades, his incisive ideas, physical principles and distinctive approach to research on numerical methods for solving hyperbolic partial differential equations have had a lasting influence on my own approach to research in the field. Here I recount some personal memories and anecdotes emerging from the rewarding experience of interacting with Phil during the Cranfield years and beyond. I also review, quite succinctly in places, some of the main research topics that he pursued throughout much of his academic career. These notes are intended as a modest homage to Roe's impact on the subject, as well as to his influence on the careers of generations of scientists in both academia and industry.

The Szemerédi–Trotter theorem, a fundamental theorem in incidence geometry, states that $n$ points and $m$ lines in $\mathbb{R}^2$ form at most $O(m+n+m^{2/3}n^{2/3})$ incidences. Recently, Lewko and independently Miao–Xie extended the Szemerédi–Trotter theorem to arbitrary fields with a correction term $O(\frac{mn}{\mathfrak{p}})$ if the field has positive characteristic $\mathfrak{p}$. In this paper, we provide some geometric intuition for the two proofs and compare them. Moreover, we show how both proofs can be improved using Hasse derivatives. As a consequence, we obtain a tight upper bound over any algebraically closed field. To be specific, when the field has positive characteristic $\mathfrak{p}$, $n\leq m$ and $\mathfrak{p}^{2(k-1)}\leq m<\mathfrak{p}^{2k}$ for some positive integer $k$, the number of incidences can be bounded by \[O\left(\mathfrak{p}^{k-1}n+m+m^{2/3}n^{2/3}+\frac{mn}{\mathfrak{p}^{k}}\right).\]

Guy Brousseau began developing the theory of situations in the 1960s, with the ambition of building an experimental epistemology of mathematics. Over the decades, the theory has been structured, consolidated and extended, drawing on the contributions of numerous researchers. It is now recognized as one of the ”genuine ground theories” in mathematics education. In this chapter, we will describe the foundations and reasons for the existence of this theory, its systemic approach to teaching and learning issues, its current architecture, which articulates a theory of mathematical situations and a theory of didactic situations, and its key concepts, as well as its functions, particularly its predictive and explanatory functions. We will illustrate this presentation with an example in geometry that also highlights the role played by the didactic engineering methodology in this theory and related work.

In this paper, we associate a canonical topology with the transition operator of a discrete-time Markov chain. We prove that this construction defines an Alexandroff topology, called the Markov Accessibility Topology, and subsequently establish that it coincides with the Alexandroff topology induced by the accessibility relation. This establishes a direct connection between probabilistic accessibility and order-theoretic topology, allowing several classical notions from Markov chain theory to be interpreted in topological terms. Among other results, we characterize absorbing subsets as closed sets, relate irreducibility to topological connectedness and the triviality of the topology, describe minimal open neighborhoods through accessibility and hitting probabilities, and investigate the communication quotient together with its relationship to lumpability.

This article is a geometric rediscovery of the singular value decomposition, with a further claim: the construction it builds is the machinery behind much of machine learning. The same argument that answers an idle question about ellipses is the algorithm behind principal component analysis, kernel methods, and PageRank, and it is not only the results that transfer but the proofs themselves, run as procedures. The usual introduction states $A = U\Sigma V^T$ and justifies it via the spectral theorem applied to $A^T A$. This is correct but unilluminating, since it assumes a powerful theorem to reach a result that is, in the end, about ellipses. Part I reverses the order. A linear map sends the unit circle to an ellipse; one asks which input directions map to its axes, and finds, example after example, that they are perpendicular. In the plane this can be watched: rotate a frame, track how far its images are from perpendicular, and a sign change forces a frame where they are exactly perpendicular, which is also where the map stretches hardest. Maximizing the stretch and recursing generalizes this to n dimensions, with singular values falling out in order, and the construction proves the spectral theorem rather than assuming it. Part II puts each construction to work: maximize-and-recurse becomes the power method and PageRank; the lemma locating the maximizer becomes the stopping rule of gradient descent; the duality between $A^T A$ and $A A^T$ becomes the transport at the heart of kernel PCA. Each connection is stated with its boundary, saying what the decomposition supplies and where another idea takes over. Prerequisites are the standard sophomore sequence, and the worked examples are small enough to check by hand.

This is a course about Ethics in Mathematics. That is, about the ethical issues that arise in mathematics itself, in mathematical work, and in the actions of mathematicians, broadly interpreted to cover all mathematically-powered technologies. This is a novel course. Traditionally, ethics has seldom been considered by, let alone taught within, the mathematics discipline. This course seeks to change that, by bringing to light the specific ethical issues relevant to mathematics, in a way that is clear and understandable to mathematicians. This course is written by mathematicians, for mathematicians, and about mathematicians. It presents new and original material, in the form of case studies covering harmful and unethical behaviour, analysis of the causal chain of events leading to such harm, and examination of the surrounding processes that drive such actions. All this is fully cited, and supported with interactive activities to aid learning. It develops much of the foundational theory of ethics in mathematics: what can go wrong, why it goes wrong, and how to address it. Students and practitioners alike take away an understanding of the context of their mathematical work, and how to change their actions to ensure more appropriate and useful outputs. In contrast to more traditional philosophy teaching, this course does not delve into the depths of ethical theory. Rather, it presents a practical, hands-on, pragmatic approach to ethics in mathematics, treating it as a branch of applied ethics. It uncovers the very particular ethical issues that arise from mathematical technologies, as well as show how general principles from ethics and responsible development apply even to mathematical work. As such, this makes it highly relevant to adjacent areas such as computer science, physics, engineering, and any other numerate discipline.

This paper investigates the preservation of three classes of Ehrhart properties under various operations on lattice polytopes. These operations include Cartesian products, lattice joins, lattice pyramids, free sums, Minkowski sums, Cayley sums, reflexive polarity, and integral dilations. Specifically, we consider the following properties frequently studied in Ehrhart theory: (i): Positivity of Ehrhart coefficients, including Ehrhart positivity and magic positivity. In particular, we present lattice point counting formulas for the polytopes generated by these operations. (ii): Coefficient properties of the $h^*$-polynomial, including symmetry, unimodality, log-concavity, ultra log-concavity, real-rootedness, and $\gamma$-positivity. (iii): Geometric properties, including the spanning property, the integer decomposition property, very ampleness, and the existence of unimodular triangulation, regular unimodular triangulation, and quadratic triangulation. We determine which properties are preserved under these eight operations, establishing preservation theorems or constructing explicit counterexamples. Furthermore, when a property is not preserved in general, we investigate sufficient or equivalent conditions for its preservation.

In his Tau Manifesto, Michael Hartl presents a formula, credited to Jeff Cornell, for the volume of the unit $n$-ball in terms of the quarter-turn constant $\eta=\pi/2$. We read the formula as a set of clues to a geometric proof, leading through orthant geometry and a higher-dimensional version of Lambert's equal-area projection. We conclude with some additional thoughts on the circle constant.

Instructional practices that require student sensemaking of mathematics are considered critical to learning outcomes, yet strategies for measurement of this instruction at scale have yet to be established. This study aims at validating and exploring the predictive validity of three student survey measures: students' opportunity to engage with, comfort engaging with, and utility of engaging with instructional practices targeting sensemaking in mathematics. Using an initial sample of 1,955 US middle school student responses, we conducted exploratory factor analysis to establish sensemaking construct structure, identifying two distinct factors reflecting students' active and passive participation in mathematical sensemaking. We then analyzed data from an additional 3,170 surveys collected over two years in the same context to confirm measurement properties and evaluate the measures' predictive validity. The resulting measures capture students' reported opportunities to engage in sensemaking practices (Sensemaking Opportunity), their comfort with engaging in those practices (Sensemaking Comfort), and their perceived utility of those practices when tackling new or difficult math problems (Sensemaking Utility). While Sensemaking Comfort and Sensemaking Utility both showed strong predictive validity over two years, Sensemaking Utility provided the most reliable estimates of student sensemaking. These findings offer researchers and practitioners a psychometrically sound, scalable toolkit for monitoring sensemaking in mathematics classrooms with implications for both large-scale research on instructional quality and teachers' ongoing efforts to monitor and improve their practice.

Euler's number $e$ is often introduced through compound interest, an infinite series, an area, or the slope of an exponential function. The last viewpoint is especially attractive: choose the base $b\in (2,3)$ so that the graph of $y=b^x$ has tangent slope $1$ at $(0,1)$. However, a curious student may raise several questions that a brief introduction necessarily leaves implicit. What does $b^x$ mean when $x$ is irrational? Why is $b^x$ differentiable at $x = 0$? How can one show that the slopes corresponding to bases $2$ and $3$ lie on opposite sides of slope $1$? Finally, why is there a unique base whose slope is exactly $1$? The paper provides an answer using tools from a first course in real analysis, including limits, continuity, derivatives, and the Intermediate Value Theorem. Its primary audience is undergraduate students familiar with standard concepts from an introductory analysis course who wish to apply what they have learned to explore questions left open in earlier courses. A secondary audience is instructors who may use selected portions as supplementary material in an introduction-to-proofs course. The mathematical ingredients are classical; our aim is to organize them into a self-contained exposition and to make explicit the logical steps that are often compressed in standard treatments.

At face value, composition operators look like something first experienced in algebra. It's just composition of functions after all. However, there is a deep and technical setting in which these objects live, behaving like infinite-dimensional matrices acting on strange yet fascinating spaces. We take a novel approach to introducing a dense mathematical topic by mirroring that of a so-called scratch kitchen, where ingredients are not prepackaged, processed, or frozen but rather are fresh. While some mathematical maturity will certainly help in the same way that baking experience helps with a new recipe, nothing in one's background is assumed. Along the way, we make no reservations about making historical or philosophical comments to spark reader's interest as well. Bon appetit!

This article proposes and articulates structure-driven methodology as an emerging cross-domain paradigm. In contrast to the objective-driven paradigm that has dominated inverse-problem solving for over two centuries, structure-driven methodology advocates identifying and exploiting the intrinsic structure of a problem to construct the solution method directly, rather than approaching the solution through iterative search. Drawing on examples from the Liouville-variant Goldbach conjecture, the three-dimensional Kakeya conjecture, Noether's theorem, and the author's nearly four decades of geophysical inversion practice, the article illustrates the common logic of structure-driven thinking in number theory, pure mathematics, physics, and seismic imaging. The author also systematically proposes the Structure-Driven Inversion (SDI) paradigm, comprising Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). Structure-driven methodology does not negate objective-driven methodology but complements it: objective-driven is a general search strategy, whereas structure-driven is a custom direct strategy. The article concludes that the significance of this shift lies not only in efficiency gains but also in cognitive deepening—when facing complex problems, the first question is no longer “how to solve the problem faster” but “what is the structure of the problem.”

In the French pedagogic tradition of Gaspard Monge, geometry was learned by drawing and handling models: drawing was knowing. The formalism of the early twentieth century placed illustration on the side of intuition, perhaps useful for discovery but kept apart from proof. Yet illustration stayed active in mathematics and provides a model for mathematical knowledge as proofs can be generated by automated systems and the Leiden Declaration asks again what mathematical knowledge is. We trace a persistent tradition of illustration through Hadamard, Coxeter, Conway, Thurston and the Geometry Center, and draw on the philosophy of mathematical practice, including Manders' distinction between exact and coexact claims, De Toffoli's account of diagrams with justificatory force, and Giardino's representational affordances. Using examples from Schwartz's minimal flat torus, early images of the Mandelbrot set, and shader renderings of algebraic starscapes, we argue for the development of rigorous illustration, to help inspire and certify the insights it affords. We close by asking the mathematical community to recognise the role illustration plays in helping develop mathematical research and understanding.

Problems developed for a mathematics circle conducted throughout the school year in a primary school class, involving all pupils without any selection, thus working with a heterogeneous group of children.

In this article I revisit the frequency interpretation of probability of von Mises, and argue that it contains the key elements of any coherent interpretation of scientific probability - notwithstanding the heavy criticisms von Mises collected from mathematicians and philosophers alike. I propose a modified definition that is not based on von Mises' "infinite collectives" but retains an essential ingredient of his interpretation, namely that probability can only quantitatively be defined for events that are, or can be, repeated in similar conditions and that exhibit frequency stabilisation. New is that the mentioned "conditions" should be "partitioned". Thus, I will partition probabilistic systems into object and environment, or in object, initiating, and probing subsystem, and show that such partitioning solves a series of classic problems. Other consistent interpretations can be unified with frequentism, e.g. Bayesianism à la Jaynes. I explore in detail the tantalising subjective or operationalist touch of probability, and argue that it can be put on objective grounds. One further upshot of the model is that several quantum puzzles can be seen as deriving from an imprecise interpretation of probability.

The growing development of online educational video resources over the past two decades, has significantly reshaped university students' self-regulated study approaches and influenced diverse ways of learning. Within an environment of abundantly and publicly available online resources, educators cannot always control which materials students choose to supplement their study. Consequently, considerable deviations are likely to be present between the taught content by the educator and an arbitrary online educational video the student watches, in terms of terminology, notation and methods, especially in mathematical sciences. These deviations can be even more considerable when Transnational Education (TNE) students incorporate native language resources to comprehend the material. This paper presents the design, creation and implementation of interactive video resources produced in alignment with established pedagogical frameworks to support an undergraduate TNE mathematics module.

Tablet-based teaching and learning has become increasingly common in university mathematics courses, offering capabilities such as digital ink, dynamic visualisation, and real-time annotation. However, there is limited experiential evidence regarding how students perceive the specific benefits and drawbacks of tablet-supported teaching and learning, particularly in Transnational Education (TNE) settings. This paper presents results from a small-scale exploratory survey of undergraduate students ($n = 19$) attending mathematics courses within an applied sciences degree programme, who anonymously rated a set of statements regarding their experiences with tablet-based instruction on a five-point Likert scale. Initial analysis shows that students in this cohort perceived tablet teaching most positively in supporting step-by-step understanding and active engagement during lectures through real-time annotation and structured note-taking. In contrast, perceptions related to long-term recall, interest in mathematics, and preference over alternative teaching methods were more mixed. These findings suggest that students in this cohort perceived tablet-based teaching positively for supporting procedural understanding, while perceptions of other aspects of learning appeared more varied.

This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research. Technological developments have repeatedly transformed the practice of mathematics. Recent artificial intelligence technologies, including symbolic and neural methods for the generation and formalization of mathematics, may already have initiated a significant chapter in this long history. Among researchers, artificial intelligence has produced a wide range of reactions: enthusiasm for its potential to yield new discoveries; intimidation by the pace of developments; indifference to these rapid changes; and concern for the implications, both for mathematics and in wider society. Mathematicians have a choice about whether and how to adopt artificial intelligence in the conduct of their research. They also have a responsibility to ensure the continued flourishing of the discipline. This Declaration calls upon mathematicians to exercise this responsibility, and provides recommendations for individuals, institutions, government, and industry. Although we adopt the perspective of mathematical research, much of what we write applies equally to other aspects of mathematics. This includes work in the broader mathematical sciences, education, mentoring, publishing, funding, science policy, and use of mathematics in the wider world.

We develop a finite-band method for the reduction of families of indefinite integrals. The starting point is the case in which the logarithmic derivative $K'(x)/K(x)$ is a rational function. This leads to adapted polynomial and Laurent-type bases and to upper triangular band matrices. The framework includes generalized real Schwarz–Christoffel integrals and more general hyperexponential weights. We then extend the construction to the case in which the logarithmic derivative $K'(x)/K(x)$ is algebraic over the field of real rational functions. The associated finite-dimensional differential module gives rise naturally to block-band systems. Finally, we show that functions satisfying linear differential equations with real polynomial coefficients give rise to finite recurrences for their moments. The resulting reduction is effective: for a prescribed target integral it is enough to solve a finite triangular band system, and only the reduction data associated with the target are required. We also discuss arithmetic complexity and give explicit examples illustrating the scalar and block reductions. The algorithms are implemented in the accompanying Mathematica package BandedReduction.