Papers

Advances in neural theorem provers have been impressive, but the successes obscure a broader vision of what AI can do for mathematics and how mathematicians can engage with AI. This essay advances a more expansive and optimistic point of view.

Book VII of Euclid's Elements represents a remarkable mathematical achievement, marking the birth of number theory. Although it falls short of explicitly stating the Fundamental Theorem of Arithmetic (that every natural number is uniquely a product of primes), it contains all the tools necessary for its proof: the Principle of the Least (equivalent to Mathematical Induction) and arithmetical anthyphairesis for finding the greatest common divisor. Our work presents novel arguments that Book VII evolved directly from early Pythagorean arithmetized music. The accounts attributing this arithmetization to Pythagoras' acoustical experiments were shown to be fictitious by Vincenzo Galilei. In contrast, the alternative experiments and Hippasus' 4-chord (bronze cylinders of heights 6, 8, 9, 12) are validated as physically correct by Euler's Law for pipes. This experimental foundation led to the arithmetization of musical intervals and to the discovery of musical (multiplicative) anthyphairesis. Applying Aristotle's Topics 158b24-29 Principle, we reconstruct how musical anthyphairesis was transferred - through an inductive ladder of ratios starting with multiple and epimoric ratios recounted by Theon of Smyrna - to its arithmetical (additive) counterpart. Furthermore, we show that the mathematical peculiarities in the definitions and proofs of Book VII find a convincing explanation only through their musical origin. Ultimately, Hippasus emerges as the pivotal figure who discovered musical anthyphairesis (which evolved into Philolaus' Fragment 6) and transferred arithmetical anthyphairesis to geometry, crucial for Pythagorean incommensurability and the principles of the Infinite and Finite.

In this article, we survey some fundamental results and recent advances in the design and analysis of Boolean bent functions from the perspective of their relation to the completed Maiorana-McFarland class.

We present a simple geometric proof of Kepler's first law. The core of the argument rests on a kinematic invariant that we call the velocity projection ratio - the ratio between the radial projection of the velocity and its projection onto a fixed direction parallel to the periapsis line. We show that the constancy of this ratio is an immediate consequence of Newton's laws, and that it leads directly to the focus-directrix property of the orbit. By establishing a direct link between planetary dynamics and the classical geometric definition of conics, our proof reveals the conic nature of the orbit as an inevitable consequence of the inverse-square law.

Permutation cycles are generally associated with undergraduate abstract algebra. This exploratory study examined whether students in Grades 5-11 could construct and use cycle representations in the context of the 15-puzzle. After a 45-minute teacher-guided lesson moving from puzzle manipulation to arrow diagrams and cycle notation, 313 students analyzed one of two new configurations - one solvable and one unsolvable - and used a supplied rule to classify it. Of these students, $78.3\%$ constructed a correct cycle representation, and $67.7\%$ both constructed the representation correctly and reached the correct classification. Cycle-construction accuracy was similar for the two configurations, but classification was less often correct for the unsolvable configuration. The findings concern immediate, supported performance rather than full understanding of permutation cycles or group theory. Nevertheless, they show that many students could use cycle representations after brief instruction and that constructing the representation and using it to reach a conclusion were separate demands. One reason to teach permutations is that, as finite, discrete, non-formulaic functions, they can extend students' experience of functions beyond familiar formulae and continuous graphs.

An essay, based on a public lecture delivered at the 2026 International Congress of Mathematicians, on how the mathematical community might respond to the arrival of artificial intelligence tools that are capable of performing research-level mathematical tasks. Rather than debating the capabilities of such tools, we condition on the hypothesis that these capabilities will arrive, and examine instead a question that is orthogonal to it: what the goals and values of mathematical research actually are. The problem-solving component of mathematics is used as a case study.

Drawing on Walter Benjamin's account of technological reproducibility and aura in relation to art, together with insights from Yehuda Rav and Ludwig Wittgenstein, this speculative essay reflects on the impact of AI systems on the practice of pure mathematics.

How should we assess whether large language models can perform mathematical invention? I argue that this question is currently underspecified: mathematical creativity is not one capacity but several mechanistically distinct modes of meaning-making - reflexive introspection on mathematical practice, analogical import from the sciences, problem-driven construction, and the bridging of distant domains - together with a further, cross-cutting distinction between meaning pursued because a pattern was observed and meaning pursued because it is strategically wanted, a distinction I develop through the case of conjecture-formation. These mechanisms are likely non-substitutable, so that competence in one does not transfer to the others. Grounding each in a historical case study and in an architecture-level account of current transformer-based systems, I suggest that today's models concentrate their competence in modes shaped by recombination and search over existing building blocks; if that description holds, the remaining modes are out of reach in principle, not just slower - though whether it holds is itself the open, empirical part. Because proof is getting cheaper as AI improves at generating it - a shift the field's own leading voices are now diagnosing - mathematical value is migrating toward the modes current systems cannot yet perform, and evaluations of AI mathematical ability should be organized around this taxonomy rather than around aggregate benchmarks that conflate it.

We recall definitions of linking numbers and Wu–Simon numbers for spatial graphs. We expose a `converse' to the Conway–Gordon–Sachs theorem (i.e. description of linking functions for embeddings $K_6\to\mathbb{R}^3$), and some results on Wu–Simon numbers. We conjecture and discuss a generalization of the Conway–Gordon–Sachs theorem to multiple linking. The exposition is based on plane diagrams, so no knowledge of spatial geometry is required.

In this paper we discuss the historical role played by Prof. Giovanni Giorgi (1871-1950) in the development of operational methods in mathematical physics. In the literature, the analysis of the scientific contributions of Giorgi is discussed in many historical papers mainly about the $MKS\Omega$-system and its contributions in the field of electrical engineering. Starting from the obituary written by Prof. Dario Graffi (1905-1990), here we analyze in detail the contributions given by Giorgi in mathematical physics, especially in the framework of the studies started by Heaviside (1850-1925) in order to obtain a symbolic representation of the solutions of differential equations emerging in physical problems. \\ Moreover, we underline the little known contribution of Giorgi to the analysis of derivatives of any real order, namely fractional derivatives, according to the present notation. \\ The main aim of this note is to underline the historical relevance of Giorgi in the mathematical foundations of operational methods in relation to the solutions of concrete problems emerging in applied physics.

What constitutes depth in a mathematical proof? This paper addresses this foundational question through a close case study of Andrew Wiles's proof of Fermat's Last Theorem. We propose a five-criteria framework: Difficulty, Originality, Fruitfulness, Unity, and Explanatory Power. By critically examining the limitations of each criterion and the intricate interrelations among them, we show that they do not merely coexist but capture genuinely complementary aspects of proof depth. Applying this framework to Wiles's proof, we then demonstrate that its celebrated depth does not reside in any single virtue, but rather emerges from a synthesis of all five criteria. Our analysis thus offers not only a novel lens for understanding Wiles's achievement, but also a flexible framework for evaluating depth across mathematical proofs more broadly.

There are two possible answers to "$(a+b)^2 = a^2 + b^2$". Either "this is completely wrong" or "I assume, you are in characteristic $2$". If we go for the first answer, we could ask ourselves: "How wrong is it actually?" In this expository article, we introduce and use resolution of singularities and the real log canonical threshold (RLCT) as a measure for such errors. The RLCT is an invariant with connections to the minimal model program and, as recently discovered and investigated, to Bayesian statistics and machine learning. Inspired by typical mistakes from high school mathematics, we compute this invariant for several classes of examples, among them the error arising from the "freshman's dream" above and from a flawed computation of means in elementary statistics. In some examples, we highlight how a computer algebra software can be used in order to solve this problem.

How many sides does a tire have: two, three, or only one? The question is a joke only until we ask what the word “side” is supposed to measure. This article follows three reasonable answers into one small piece of topology: a loop on a tire can have no endpoints and still be a border of a chosen region. The point is not to find the official number of sides, but to see why the count changes when the model changes.

Large language models have begun refuting long-standing conjectures and solving long-open problems. The introspection this has prompted about the future of mathematical discovery is well under way, and the anxiety accompanying it legitimate – but both, we claim, are attached to the wrong loss. What machines now produce is the countable part of mathematics – theorems, proofs, refutations – which was always the work's residue, not its product. The distinction is old, and not economic: a result can be taken in its finished essence, or in the operations that engendered it. 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: understanding tells us where to look; looking produces the residue; and taking it up again, one journey at a time, rebuilds 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. A new instrument makes a new way of working before it makes a new result. The pressing questions are therefore institutional: who can check an announced result, whoever announces it; what work and training become for the next generation of researchers; and whether the one thing that cannot be mass-produced – the journey that nourishes a shared understanding – continues to be funded. Mathematics, we argue, is uniquely placed among the sciences on the first – a proof answers to no one's permission – and uniquely exposed on the other two: teaching cannot go on as before, and no collective position yet exists; and the journey has never had a price our institutions knew how to pay. The decision is ours.

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.

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 $M_{127} := 2^{127} -1$ 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.

In this paper, we give a memorial tribute to Prof. Erik M. Bollt (1967–2025).

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.

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.

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.