Papers

Artificial intelligence threatens to outrun human understanding and control. New mathematics is needed to design AI that is legible, steerable, and cooperative with humanity. I organize this invitation by mathematical field, so you can turn straight to your own: logic and game theory for cooperation; probability for agency and world-models; algebra and representation theory for learned features; analysis and geometry for generalization and training dynamics. Each section ends with an open problem that is accessible to a working mathematician with no prior experience in AI safety.

We study self-referential multiple-choice tests in which the answer choices are positive integers and the question asks how many answer choices are correct. These tests quickly lead to several kinds of ambiguity and paradox. The paper presents the ideas through a dialogue between Queen Tanya and her ten sages, developing a framework for studying these immensely questionable tests.

What can a finished text reveal about the process that produced it? Drawing on Turing's imitation game and statistical decision theory, this article examines the limits of AI-text detection as an inference from a completed object to an unobserved history. For two known source distributions with equal prior probabilities, a standard identity gives the minimum average classification error as half their probability overlap. This overlap is one minus their total variation distance. Perfect detection therefore requires nonoverlapping distributions; useful discrimination does not. In practice, the problem is harder: human and machine writing form changing families of distributions, posterior probabilities depend on base rates, and “AI-written” becomes ambiguous when people and software contribute to the same text. Turing's interrogator can ask another question; a detector restricted to finished prose cannot. Distinguishability, source attribution, authentication, and compliance are different inferential tasks. The paper's own documented human-AI provenance shows what a production history can reveal that a binary label cannot.

Starting from a Hilbert plane and removing the Side-Angle-Side (SAS) congruence axiom, we investigate to what extent SAS can be recovered synthetically from the remaining classical triangle congruence criteria. We show that the Angle-Side-Angle criterion, together with a ray correspondence principle corresponding to Theorem 13 of Hilbert's Grundlagen der Geometrie, suffices to reconstruct SAS. We further show that both the Side-Side-Side and the Side-Angle-Angle criteria also suffice, once combined with the ray correspondence principle and suitable auxiliary principles – the existence of midpoints and a hypotenuse-angle criterion for right triangles in the first case, and the existence of angle bisectors, the congruence of supplements of congruent angles, and the Pons Asinorum in the second. Although the two routes rely on auxiliary principles of different character, we show that they converge on a single final argument once a common hypotenuse-angle criterion is established. A metamathematical analysis, based on an explicit model adapted from Hilbert's own independence construction, complements these reconstructions: it shows that the ray correspondence principle alone cannot reconstruct any of the classical criteria, and that the Pons Asinorum and the hypotenuse-angle criterion are each independent of the remaining auxiliary principles used in their respective reconstructions. The resulting picture is not a formal hierarchy of the congruence criteria, but it does show that the Angle-Side-Angle reconstruction rests on a provably more economical basis than those obtained from Side-Side-Side or Side-Angle-Angle.

We report on a Lean 4 formalization of Sullivan's No Wandering Domains theorem: every Fatou component of a rational self-map of the Riemann sphere of degree at least two is eventually periodic. The project formalizes the relevant background in normal families, the Montel-Carathéodory theorem, Julia and Fatou sets, local Sobolev regularity and Wirtinger derivatives, the Cauchy and Beurling transforms, the equivalence of analytic and geometric quasiconformality, and the measurable Riemann mapping theorem. This paper describes the translation of this mathematics into Lean, the proof architecture, the reusable components, and the autoformalization workflow.

The problem of the Nile has been stated by R. Fisher in 1936. This problem concerns the fair division of an agricultural land and is the eponymous problem of a general class of problems on measures and similar regions. In 1938, W. Feller has shown that the general problem has no solution. However, we will show that under a natural hypothesis, the original problem of the Nile is solvable with connected plots.

This is a survey on an old open problem in combinatorics called the unit distance problem, and the field of mathematics around it, called incidence geometry. What do we know about the problem? Why is it difficult? How does it connect with other parts of math?

The Portmanteau Theorem gives several equivalent characterizations of weak convergence of probability measures (equivalently, convergence in distribution of random variables). This note presents a unified proof of the theorem in a seven-statement formulation, emphasizing the structure of the implication cycle connecting expectations of bounded continuous functions with probability bounds on open and closed sets. The argument highlights standard approximation techniques and clarifies the role of intermediate function classes such as bounded Lipschitz functions.

OpenAI recently announced a proof of the Cycle Double Cover (CDC) Conjecture. Most media reports have characterized it as a 50-year-old open problem. In reality, according to a 1987 letter from Tutte to Fleischner, the Cycle Double Cover Problem has been open for at least 80 years. Two early results regarding the CDC conjecture were established in one of Tutte's 1949 publications.

A categorial approach to the non-metric geometry in Books V and VI of Euclid's Elements is presented. Specifically, we introduce a diagrammatic syntax that can be overlaid immediately on his diagrams, thus bridging intuitive presentation with fidelity to Euclid's arguments. This syntax makes complicated definitions like V.5, and indeed the arguments throughout books V and VI, including arguments about similar figures, intuitively clear. We show in an appendix that this syntax can be used to solve a puzzle regarding ancient mathematics. Finally, we offer evidence that this approach to Euclidean diagrams is rooted in the Aristotelian tradition itself, and that a similar syntax was utilized, in antiquity, for related questions of numeric and proportions. Thus the syntax is plausibly faithful to Euclid's own thought-world, and not an outside-imposition.

We propose a general approach to enharmonicism within syntactic music theory; that is, we formalize several abstract structures of enharmonicism without any acoustic considerations. The structures of concern to this paper are constructed from diatonics, chromatics, and enharmonics, and we seek to present a general enharmonic theory that builds upon Hook's enharmonic theory published in "Enharmonic Systems: A Theory of Key Signatures, Enharmonic Equivalence and Diatonicism" (2007). We also propose that the two criteria of reflection and maximal evenness can be used to determine whether an arbitrary enharmonic system is musically practical. Moreover, we argue that preferential treatment of the standard enharmonic system (SES) can be motivated from an abstract mathematical perspective divorced from pitch consideration and conventional acoustic constructions of SES.

This book develops algebraic and combinatorial methods for studying discrete structures. It brings together graph theory, Boolean functions, Fourier analysis on finite groups, coding theory, perfect colorings and perfect codes, association schemes, Latin squares, and related topics. A central theme is the interaction between different representations of the same object: combinatorial, algebraic, spectral, and coding-theoretic. The main mathematical object studied in this book is a perfect coloring of a graph, or, equivalently, an equitable partition of a graph. The book is intended for advanced undergraduate and graduate students in mathematics and computer science, as well as for researchers in discrete mathematics, combinatorics, coding theory, and related fields.

We analyze the poetico-comedic criteria by which Tom Lehrer arranged the chemical elements in his 1959 hit "The Elements," a parody to the tune of Gilbert and Sullivan's "The Modern Major-General" (1879). We compose a new version to the same tune incorporating all 118 currently known elements and fulfilling, so far as possible, the same artistic aims.

We present some basic recommendations for (mostly graduate) students when it comes to choosing to use, or avoid, AI in their mathematical research and education. These include: maintaining a connection to the tried and true ways for learning mathematics; focusing on communication and the sharing of ideas with other people; thinking critically about the politics of labor and organizing with your cohort and others at your home institution; and being in touch with the realities of the impact of AI infrastructure on land and people.

Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poincaré disk can obscure. To bring hyperbolic geometry to life, we embed hyperbolic surfaces in Euclidean space by discretizing the surfaces into meshes, and minimizing a distortion energy so that the edge lengths in the embeddings match those in the hyperbolic plane. The resulting surfaces buckle and ruffle to accommodate the extra area, making visible what flat models hide. We present exemplary illustrations, such as embedded disks, equidistant strips, diverging geodesics, and also artistic organic-like renders. We discuss our use of these models, as renders and 3D prints, in research talks, public engagement, outreach, and education.

We present a machine-checked Lean~4 formalization of Dong and Yang's classification of optimal finite-length $(n,4)$ binary block codes for binary symmetric channels. The formalization was developed mainly by feeding the paper's proofs to an AI tool. To establish correctness, the authors verified the main theorem statements in Lean and the accepted axioms. This note discusses the corrections and simplifications made to the AI-generated formalization, and records discrepancies found in the paper during the formalization. The Lean code is available at https://github.com/shhyang/n4code_lean.

Let $n\geq 1$ be an odd integer, set $m=(n-1)/2$ and let $M$ be an $n\times n$ matrix whose coefficients are of the form $M_{i,j}=aij+bi+cj+d$ where $0\leq i,j\leq n-1$. Then we prove that for all squares centered at the central coefficient $M_{m,m}$, the mean value of the square equals $M_{m,m}$.

The way numbers are represented strongly influences which arithmetic structures are easy to see. The prime clockwork is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit mod or div operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=\nu_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this grand cycle is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

The Fibonacci and Lucas sequences are old friends in recreational number theory. Here we ask a simple question: which of their terms $F_n$ and $L_n$ are Smith numbers? A Smith number is a composite integer whose decimal digit sum equals the sum of the decimal digits of its prime factors, counted with multiplicity. The question is easy to state, but for large $n$ it quickly becomes a factorization problem. Using available complete factorization data, we carried out computational searches in both sequences and obtained new Smith terms in each of them. For the Lucas sequence, the first author's search found ten Smith terms with indices below 1000: 3, 95, 105, 114, 183, 437, 609, 682, 827, 902. These results were published in the On-Line Encyclopedia of Integer Sequences as OEIS A395686. Four further indices, 1090, 1153, 1215, 1378, were subsequently added to the sequence by Sean A. Irvine. For the Fibonacci sequence, starting from the cases already recorded in OEIS A382922, our search produced seven further Smith numbers: $F_{1440}$, $F_{1554}$, $F_{1596}$, $F_{1863}$, $F_{2256}$, $F_{2277}$, $F_{2559}$. For the even-index cases, the identity $F_{2n}=F_nL_n$ allows available Fibonacci and Lucas factorization data to be combined to recover the complete factorizations needed for the Smith test. Surprisingly, the two searches also meet at index 827: both $F_{827}$ and $L_{827}$ are Smith numbers.

Plato's Parmenides has long been considered the most difficult and enigmatic of Platonic dialogues, marked by seemingly contradictory arguments. We present an interpretation showing that the One of the Second Hypothesis (142b-155e) is an intelligible Being whose structure constitutes the philosophical analogue of a dyad in periodic anthyphairesis. First, by linguistic analysis and Proclus' Platonic Theology, we show that the One and the Being form an indefinite dyad satisfying the philosophical analogue of infinite anthyphairesis. We analyze Plato's initial introduction and eventual rejection of eristic numbers formed with unequal units from the infinite sequence of remainders (143c-144c). Second, we reveal that the presence of the One in the Being (144c-d, 138a) is achieved via contacts ("hapseis", 148d-149d). Based on the Pythagorean musical rule relating terms and intervals, a contact is identified with the ratio between consecutive anthyphairetic remainders. The circularity of contacts (138a) expresses Theaetetus' Logos Criterion for anthyphairetic periodicity, defining dialectical numbers with equalized units and establishing the finiteness of the dialectical number of parts. This framework resolves key Platonic enigmas: Plato's Indivisible Line is identified with the intelligible Being; the Sophist's assertion that "not-Being is a Being" corresponds to the equalization of the dyad; and the compresence of opposite properties separates intelligibles from sensibles. Finally, the self-similar Oneness of an intelligible Being refutes Vlastos' Non-Identity principle, resolving the Third Man Argument.

We discuss the Babylonian division of trapezoids and the construction of such trapezoids with rational sides and transversals from Pythagorean triples.

As several mathematical conjectures have recently been settled using large language models (LLMs), the mathematical community has formulated norms and recommendations regarding the publishing of such results. These norms do not cover the disclosure of the prompts and precise software setup used to obtain those results, nor do they require that results be formalized in a manner that allows for machine verification. I argue that both of these are essential. In addition, since LLM-obtained results may be hard to understand, human authors have the responsibility to invent intuitive explanations.

Mohsen Hachtroudi (1908–1976) was one of the founders of modern mathematics in Iran and the first Iranian mathematician to gain international recognition for original research in modern times. A student of Élie Cartan, he introduced the canonical projective connection now known as the Hachtroudi connection, whose significance has become increasingly apparent through later developments in Cartan geometry, CR geometry, and parabolic geometry. This article combines a historical and mathematical study of\\ Hachtroudi's life and work. It examines his education in Iran and France, his role in establishing modern mathematical research and higher education in Iran, and his influence as a teacher, essayist, and public intellectual. It also presents a modern account of his mathematical contributions and their continuing impact on the geometry of differential equations and the theory of geometric structures.

I propose that mathematicians who use AI in research maintain a research profile: a record of what the system has learned about working with them, where those lessons came from, and whether they remain useful. I describe how such a profile grew out of my own mathematical work, through corrections, failed approaches, and repeated attempts to keep difficult ideas from drifting toward easier ones. The profile is not a portrait of the mathematician and does not replace mathematical judgment. It belongs under the mathematician's control and must remain open to revision. I invite others to develop profiles of their own and report what they learn.

The Egyptian knotted rope has been historically considered a measuring/geometric tool. A naive change in the way the rope is held to tighten it turns the rope into a primitive calculating tool — a sort of primitive abacus. The arithmetic skills of the knotted rope have a double interest. From an historical point of view, they fit perfectly with some of the basic ancient Egyptian computing techniques. From an educational point of view, they enhance the potential interest of the knotted rope as a useful manipulative in mathematical education.

Hitomezashi is a particular style of sashiko stitching characterized by crossing lines of stitches. Certain hitomezashi designs can be encoded with two binary strings and have been the subject of a wide-array of mathematical research. Dubbed generalized hitomezashi patterns (GHPs) by Sen & Martinez, these designs are reversible and can be 'self-dual.' We investigate the possible 'dual-' or 'flip-symmetries' in GHPs; the inclusion of these symmetries is equivalent to considering two-colour symmetry groups. Following the work of Sen & Martinez, who found the wallpaper symmetry groups that are compatible with GHPs, this paper investigates the possible two-colour symmetries compatible with GHPs. We prove that there are four possible rosette two-colour symmetry types, completely describe the properties on binary strings that create these symmetries, and prove that exactly 17 of the 46 two-colour wallpaper groups are compatible with GHPs.

Thousands of sangaku tablets, inscribed with colorful geometry problems, were posted in shrines and temples across the Japanese archipelago starting in the Edo period (1603–1868), representing an intersection of artistic, religious, and mathematical traditions. The Sangaku Archive is an attempt to carefully document these objects using the latest digital tools, and to tell an unusual and compelling story about mathematics—one that enriches computations and proofs with historical context and cultural appreciation.

In this essay I reflect on the status of three of the Millennium Prize problems: the Riemann Hypothesis, P vs. NP, and the solvability of the Navier-Stokes equations.

Recent advances in artificial intelligence (AI) have sparked growing interest in its use for mathematical research. While some view this as a major opportunity for discovery, others have raised concerns about its impact on traditional research practices. Despite extensive debate, empirical evidence on how AI is actually being used in mathematics remains limited. To address this gap, we collected all 32,944 arXiv submissions posted between March 1 and August 20, 2026, whose primary or secondary categories included Mathematics. We identified 3,575 submissions that explicitly disclosed author use of AI, of which 1,712 involved at least one substantive mathematical contribution. Our analysis reveals several broad patterns. First, disclosed AI use increased sharply over the study period, with substantive use growing from 1.39% of Mathematics submissions in March to 14.09% through August 20. Second, substantive AI use is highly uneven across fields: Combinatorics has the largest number of such papers, while Metric Geometry has the highest substantive-use rate. Third, substantive AI use is geographically concentrated: under weighted author counts, the United States and China together account for about two-thirds of the recognized country weight. Fourth, AI is already being applied to open research problems: among 717 named open-problem records associated with substantive use, 71% are labeled as fully resolved based on the authors' descriptions, with proofs of the conjectured statement more common than counterexamples or disproofs. Finally, AI-system use is also highly concentrated, with OpenAI systems appearing most frequently, followed by Anthropic. Together, these findings suggest that AI-assisted mathematics is expanding rapidly but remains at an early and uneven stage of adoption.

Lobachevsky's discovery of Non-Euclidean geometry was a byproduct of his more ambitious plan of reforming the traditional Euclid-style foundations of geometry. This project aligned with d'Alembert's and Condillac's practically-oriented visions of mathematics and aimed at closing the perceived gap between the pure mathematics, on the one hand, and the natural sciences and engineering, on the other hand. Following d'Alembert Lobachevsky deliberately avoided the axiomatic development of his geometrical theory and combined instead some preliminary intuitive synthetic constructions with their rigorous analytic treatment. In addition to Non-Euclidean geometry, Lobachevsky's project brought about an early form of Dimension theory. The paper comprises a description of the local context of Lobachevsky's works, including some pointers to Russian geometry textbooks of the beginning of the 19th century. We summarise Lobachevsky's philosophical views which motivated his mathematical achievements and show how these motivations were misunderstood and misinterpreted in the beginning of the 20th century by Henri Poincaré and Ernst Cassirer. Finally we argue that Lobachevsky's epistemic views remain pertinent in the context of the ongoing discussion about the reunion of mathematics and physics. The Appendix comprises English translations of eight Lobachevsky's documents of various length and character where he expresses his general epistemic views on mathematics and, in particular, geometry.