Papers

We study the elementary continued fractions of Ramanujan. The entries considered are precursors of his hypergeometric and $q$-hypergeometric continued fractions. They appear in the first third of Chapter 12 of Ramanujan's second notebook.

We prove a conjecture proposed by Robert J. Lang over 20 years ago, stating that in a circle-packed origami design, any axial polygon with 5 or more sides can be reduced by properly adding a new circle. The proof is based on the structure of the medial axis of the region enclosed by the circles, and yields a concrete algorithm for finding the added circle. The result extends to configurations involving rivers.

This is a personal essay about how I see math, what I value about it, and what it means to me.

The purpose of this article is to explicitly describe how the integration technique known as the change of variables is used in graduate-level Probability and Statistics courses. Throughout the article, we employ this integration technique seven times across various examples and results. We demonstrate that four applications of this technique are required to derive the probability density function (PDF) of the Student's t-distribution.

In this time of uncertainty and anxiety about artificial intelligence (AI) in mathematics, it is more important than ever to reflect on how we can prepare the next generation of professional research mathematicians for success. In this article, we report, from the front lines, on how AI is affecting the long-running summer math research program at the University of Minnesota Duluth. We do not have answers to the big questions about how AI will shape the future of professional mathematics. What we can offer instead is a collection of practical observations about the role of AI in the 2026 Duluth program: how it was useful and where it presented potential pitfalls; what surprised us and how we adapted; and what went well and what we wished we had done differently.

Over three centuries ago our Enlightenment forefathers brought forth in Europe–and later on this continent–a new approach to knowledge production and dissemination, conceived in distrust of authority, and dedicated to the proposition that truth is accessible to human reason. Mathematics, the Enlightenment's firstborn, was its existence proof–knowledge that compels assent without appeal to authority–because the only way to make a theorem certain was to show it to a human mind. Now we are engaged in an existential struggle testing whether mathematics, or any discipline so conceived and so dedicated, can long endure. We are met in the wake of the OpenAI press release: 166 pages, 616,000 lines of Lean, produced in 88 hours by some 10,000 software agents, and read in full–at the time of this Siren Call–by no human being. A proof has historically been two things: a certificate that a theorem is true, and a demonstration that lets another person see why. OpenAI delivered the first without the second. But, in a larger sense, what matters is not what the Silicon Leviathan (OpenAI and its peer competitors) produces but rather what we accept. By accepting the certificate without the demonstration, we adopt the Hobbesian bargain: authority, not truth, makes the law. The age that this augurs is–in Panofsky's phrase–a Middle Ages in reverse–its authority a subhuman superintelligence. The great task before us is to frame a covenant under which what has become optional–understanding–remains required: that nothing counts as mathematics until a human being has understood it–and can show the next person why. Whether we will safeguard and keep our sovereign inheritance–decided at the moment of acceptance–which is not yet past–will determine if this Aufklärungsdämmerung spells the Enlightenment's end.

Lecture notes from a graduate course given by Michael Krivelevich at the School of Mathematical Sciences of Tel Aviv University in the spring semester of 2026. Topics covered include: phase transition and the giant component in $G(n,p)$; long paths and cycles in supercritical and sparse random graphs; thresholds for connectedness and perfect matching; general model of a random subgraph of a finite graph; phase transition and the giant component in the random hypercube; polynomial diameter of the giant component; perfect matchings in the random hypercube.

Powerful artificial intelligence is weakening the traditional relationship between mathematical output and evidence of mathematical expertise. In particular, the production of an original theorem or a polished dissertation can no longer, by itself, certify the intellectual formation of its nominal author. I argue that graduate mathematical education should therefore be organized around the formation of intellectual agency: internal technical competence, mathematical judgment, understanding, and responsible participation in a shared intellectual culture. I distinguish productive from premature cognitive offloading, propose complementary independent and AI-augmented modes of training, and suggest a corresponding reformulation of the role of the PhD thesis and dissertation defense. More broadly, I argue that academic mathematics should understand itself increasingly as an institution for the reproduction and stewardship of human mathematical expertise rather than primarily as a mechanism for producing theorems.

What distinguishes the solution of a difficult mathematical problem from a conceptual revolution? Some mathematical innovations do more than establish new results: they reorganize the relations among problems. We propose to describe such events through changes in the transfer structure of mathematics, namely the pattern determining which problems can naturally inform one another, which methods can move between them, and which new questions become accessible. The eighteenth-century controversy over the vibrating string provides our central historical case. D'Alembert found a famous model for wave propagation, Bernoulli proposed a modal interpretation, Euler broadened the class of admissible profiles, and Lagrange constructed a finite-dimensional strategy based on modal decomposition. Later, Fourier analysis gave the trigonometric language in which all the previous descriptions can interact. This motivates our Rewiring Thesis: a conceptual revolution is a persistent, large-scale reorganization of transfer relations among mathematical problems. The productive representation is one mechanism capable of producing such rewiring, but some stress tests (involving Galois, Riemann, Lebesgue-Schwartz and Wiles) show that it is not the only one. The musical notation provides an independent case in which the representation becomes operationally generative without being informationally complete. The resulting framework suggests a stronger criterion for artificial mathematical creativity than problem solving alone. The relevant question is not merely "whether AI can solve a famous conjecture", but "whether it can recognize that an inherited conceptual organization is itself an obstruction and produce a new one that reorganizes several problems at once". In this sense, the decisive achievement may be the invention of "the next Fourier transform".

The is an expanded version of a memorial article that is to appear in the Notices of the American Mathematical Society.

Reverse mathematics is a branch of mathematical logic dedicated to determining the minimal set existence principles necessary and sufficient to derive ordinary mathematical theorems about concrete structures like the real line. Since the mid-1970s, reverse mathematics has developed a systematic classification of the strength of theorems in areas of mathematics ranging from real and complex analysis to infinitary combinatorics. This essay will place reverse mathematics in its historical and philosophical context, and reveal its relevance to central issues in the philosophy of mathematics, from the foundational programmes of Hilbert and Brouwer to contemporary debates about realism, determinacy, and applicability of mathematics. In doing so, it will discuss the role of computability theory in measuring the strength of set existence principles, as well as related questions about idealisation when these principles are applied in the physical sciences and in philosophy.

We provide a clearer presentation of the Chair44 monotile, a new strongly aperiodic monotile in three-dimensional Euclidean space $E^3$.

We introduce the Endless Exam, a benchmark spanning fourteen parameterised families of mathematical construction problems, with verifiable scores that distinguish progress before and beyond published mathematical frontiers. Each submitted object is checked automatically for validity and assigned a relative quality score against a published frontier or construction baseline, without capping improvements at 1. The benchmark draws long-term challenges from open mathematical problems and generates larger instances by varying their parameters. Compact certificates allow large constructions to be verified without listing every element. Across nine models evaluated on 69 distinct instances, continuous quality scores distinguish performance even though none of the 30 published-frontier references is surpassed. Size-quality curves show how construction quality changes as problem size increases. We release the generators, verifiers, references, model responses and analysis to support continued measurement before and beyond human frontiers.

This article describes our experiences in mathematical outreach over the past decade. We talk about specific activities, but also general principles that we've learned along the way.

Lagrangian duality is fundamental to optimization, from the development of solver techniques to the interpretation of solutions; yet its geometric intuition can be difficult to grasp through mathematical derivation alone. This tutorial explains duality and its implications in a highly visual way for a broad audience of students and practitioners. It develops the motivation for the Lagrangian, the two-player game interpretation, and the connection to Fenchel duality. Throughout, it emphasizes the geometric meaning of the solution from both the primal and dual

Inspired by the polymath Abraham Ibn Ezra (1089-1167) who approximated {\pi} by 20/9 times the square-root of 2, and contemporary scholar, RSJ Reddy, who gave (14-sqrt(2))/4, we point out that both of these approximations are optimal in some sense, and we generate many other ones that are even more optimal, albeit not as nice.

The determinant is traditionally introduced through permutation expansions, multilinearity, or recursive expansion formulas - approaches that, while rigorous, often obscure its conceptual significance. Instead, an approach is advocated wherein the determinant is defined via its natural characterization as a multiplicatice map on matrices, reflecting its role as a structural invariant of linear transformations under composition. This perspective defers combinatorial machinery until after the core properties are estaqblished, thereby lowering the cognitive barrier for students encountering the concept for the first time. The equivalence of this characterization with classical formulations is shown, and it is argued that this framework better aligns with the structural emphasis of m odern mathematical education.

Artificial intelligence can increasingly produce plausible, sophisticated mathematical material faster than a developing graduate student can understand or verify it. A sophisticated result or paper draft therefore becomes weaker evidence of the student's own mathematical development. This creates a formation gap between output and personal capacity, and a trust gap between a convincing argument and warranted acceptance. The formation gap can persist even when the student understands the output: understanding a supplied argument does not by itself establish the capacity to initiate and direct inquiry. These gaps are not the whole story. AI can also help students explore examples, compare approaches, enter unfamiliar areas, and undertake ambitious research. The task is to design an apprenticeship that realizes these possibilities while developing substantive mathematical command. The central purpose of a mathematics PhD is to form mathematicians capable of original, independent, and responsible inquiry, including inquiry conducted with AI. This document develops that objective through four connected capacities: competence, judgment, independence, and responsibility. It distinguishes a work's contribution to mathematics from the evidence it provides of a student's formation; explains how a known answer can initiate rather than end creative inquiry; and proposes changes in learning activities, assessment, doctoral originality, advising, and institutional support. Purposeful independent work and ambitious AI-assisted research are complementary parts of the model. Its recommendations include proportionate contribution statements, recognition of advising costs, and staged pilots evaluating both mathematical ability and effective human–AI collaboration. The aim is not to preserve an inherited sequence of training, but to improve mathematical formation as mathematical practice changes.

Gödelian incompleteness yields arithmetical sentences $A$ such that $PA+A$ and $PA+\neg A$ are both consistent. Are such extensions equally legitimate? I propose graded epistemic arithmetical non-pluralism: justification for choosing between them varies with the set-theoretic strength of $A$. I defend $PA+Con(PA)$ and show Koellner's non-pluralism for first-order arithmetic is inadequate given Friedman's concrete incompleteness. Resolving the selection problem for such sentences turns on justifying large cardinals. Defending graded non-pluralism thus engages Gödel's programme and the justification of very large cardinals.

We present reflections on an experimental seminar studying the human and cultural roots of modern mathematics.

In this article, on Descartes' famous "circle theorem," we first attempt to explain how the philosopher could have arrived at the equation that he presents without demonstration in a letter to Princess Palatine Elizabeth of Bohemia in November 1643 (a previously unpublished proof, to our knowledge). Then we report some stages of the subsequent generalization of this theorem, whose Cliffordian flavor, via the equivalence of a quadratic form and the square of a linear form, still mobilizes mathematicians today. This statement, which, over time, has undergone different extensions, Euclidean and non-Euclidean, has found, in fine, a spinor formalization. We see there the proof of what we may call, in a Bachelardian style, an "inductive value" of the truth, which extends by successive generalizations. We conclude, more briefly, with the contacts between Elizabeth and Descartes, and the perhaps symbolic meaning of these considerations on "kissing circles," as they are called in Anglo-Saxon countries, in the context of their correspondence on the soul and the body, and the question of passions.

In the first part of this work we showed that, within a Hilbert plane deprived of the Side-Angle-Side axiom, the Side-Angle-Angle criterion, together with a ray correspondence principle [RCT], the existence of angle bisectors [AB], the congruence of supplements of congruent angles [SA], and the Pons Asinorum [PA], suffices to reconstruct SAS. We left open the question of whether [PA] is genuinely required alongside the other three principles, noting only a qualitative asymmetry in the nature of the principles involved. In this second part we answer this question: we show that \begin{equation*} SAA,\ [RCT],\ [AB] \;\vdash\; [PA], \end{equation*} so that [PA] is redundant among the hypotheses of our main theorem, which improves to \begin{equation*} SAA,\ [RCT],\ [AB],\ [SA] \;\vdash\; SAS. \end{equation*} The proof adapts an argument recently given by Donnelly, who reconstructs SAS from SAA together with an angle addition axiom and the existence of angle bisectors.

I believe that the development of AI tools will lead to a transformation of mathematics, both professionally and culturally. However, it will be a tragic missed opportunity if the massive shifts in culture and practice that are underway are motivated and influenced solely by AI. We must aspire to more than this. We can and should collaboratively strive for a future where our cultures, practices, and values are grounded first and foremost in empathy, compassion, justice, and humanity. In this article, I call to action every member of the mathematical community, whether you view yourself as a researcher, teacher, scholar, student, professional, hobbyist, or a combination of these and more.

Generative AI changes the conditions under which graduate mathematics is learned, assessed, written, and defended. The central claim of this paper is that mathematics graduate programs should respond to the moment by clarifying what graduate mathematics education is trying to teach and assess. In most ways, the goals of mathematics education have not changed. Rather, with changing tools it has become more essential than ever to make clear the goals of mathematical training. We use the term mathematical judgment to refer to the capacity to evaluate mathematics (e.g., claims, definitions, examples, proofs, analogies, computations, uses of tools, research directions) as mathematically sound, useful, well-posed, and appropriately justified. The recommendation is to center training on mathematical judgment, and we examine possible policies for graduate programs in Mathematics to this end.

In 1878, Pafnuty Chebyshev presented to the Association fran\c caise pour l'avan\-cement des sciences [French Association for the Advancement of the Sciences] an article \cite{Chebyshev1878} dealing with garment cutting. According to Chebyshev himself, his interest was sparked by a lecture given by Édouard Lucas that he had attended in 1876 \cite{Lucas1876}. There is a second story on the origin of Chebyshev's interest in garment cutting according to which in the 1850s, being short of money, Chebyshev got himself a job as a consultant to a clothing factory. At the time of the Crimean War (1853-1856), there was a great demand for uniforms. Chebyshev was allegedly asked to optimize the use of fabric, and it was there that his interest in garment cutting was born. This second story appears to have its origin in a post by Clive J. Grant to MacTutor in 1996 \cite{Grant1996}. However, this contribution contains no references, and no other source of information that I have found offers any first-hand documentation to support this story. Our conclusion is that this second story is a fabrication, invented out of whole cloth.

In 1854 Riemann presented his Habilitationsvortrag Über die Hypothesen welche der Geometrie zu Grunde liegen to the Faculty of Philosophy of the University of Göttingen. The lecture does not contain analytical developments and Riemann never explained the reason for this choice. According to Dedekind this choice was done in order to make the lecture understandable to the non-mathematician professors of the Faculty. This is the suppression thesis. Nevertheless, in 1851, three years before, Riemann has presented his Inauguraldissertation Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse to the same professors of the Faculty of Philosophy. The Inauguraldissertation contains many analytical developments but, on that occasion, they were not suppressed. Therefore, the presence of non-mathematician professors is unrelated with the presence or absence of analytical developments in the lectures. The ideas contained in Riemann's Habilitationsvortrag –manifold, line element, curvature– were so new at the time that there was not even a proper language to express them. Therefore, Riemann was forced to adopt a non-technical expository style. This explanation for the absence of analytical developments in Riemann's Habilitationsvortrag is more credible than the suppression thesis.

From the manifest of Andre Breton, through the transdisciplinary understanding, we arrive at a post-modern manifest. A talk by Laura De Marco (Harvard) will provide scientific background to approach an AMS poetry. The next section will be a qualitative analysis of some new operations on the real numbers. The conclusions will be given in the last section, and an appendix will recall some previous work with some new comments.

We develop an interdisciplinary framework for evaluating the epistemic robustness of social-scientific inquiry through Hadamard's criteria for well-posed problems: existence, uniqueness, and stability. Reinterpreting these criteria not as demands for deterministic certainty but as methodological guardrails, we show how they diagnose recurrent failures of identification and inference across quantitative and qualitative paradigms. In quantitative research (e.g., econometric modeling), instability manifests when substantive conclusions depend sensitively on model specifications or data filtering. In interpretivist qualitative research, non-uniqueness and non-falsifiability arise when theoretical frameworks are elastic enough to accommodate contradictory observations without pre-specified rejection criteria. We frame these failures as inverse problems where the mapping from empirical data to substantive claims fails to satisfy existence, uniqueness, or stability. Finally, we propose cross-paradigmatic safeguards: ex-ante falsification criteria, empirical boundary conditions, multiverse sensitivity auditing, and cross-observer validation, to ensure social-scientific claims remain appropriately constrained, falsifiable, and robust to perturbations in evidence and interpretation.

Motivations and derivations for Heron's triangle area formula abound. But how would one discover the formula? We recount the challenge of designing a laboratory experiment to discover Heron's formula, and of doing so in the morning, in time for a noon class—a snapshot of collaboration with AI in fall 2025. We then reflect on extensions and generalizations.

We present an exposition of a proof, discovered by GPT-6 Astra, of the Erdős–Sós Conjecture, which states that every graph with average degree greater than $t-2$ contains every tree on $t\geq 2$ vertices.