cube |=> cube(cube)

holey, holey, holey

Two months ago, when I wrote about solving my Frontier Math Tier 4 problem, I did not expect the landscape to shift this quickly. Computational arithmetic algebraic geometry is turning into an incredible hotbed of ideas. This area, shaped by deep questions around elliptic curves, algebraic numbers, varieties, and the work of people like Brian Birch, Jean-Pierre Serre, and many others, has always had a strong computational undercurrent. But what is happening now feels different. The agents I have been testing, especially Codex, are reaching a level where they often outpace my own ability to write code quickly and effectively. At the same time, I can still curate, inspect, redirect, and judge the mathematics. That combination is extremely powerful. I can jump into almost any algorithm I need, optimize it, decompose it, rebuild it, and move between Magma, SageMath, and Rust with a kind of flexibility that still feels unreal. This is not "vibe coding". It is extreme engineering guided by mathematical taste. In my recent projects, this has already helped me close two big questions. The technical conversations I can now have with Codex about Magma code, computational algebra, and arithmetic geometry are honestly stunning. Big computational problems in arithmetic geometry are going to fall much sooner than many people expect.

james joyce would like a word... or rather many words & many worlds...

Giving a talk in the @Stanford SCIEN seminar this Wednesday (1/3) at 4:30pm: scien.stanford.edu/index.php… The topic is “normal coordinates”: a shape representation little-used outside of mathematics—but which turns out to have nice applications in geometry processing & learning.

preparing for the future of mathematics

I think one of the roles we play as mathematicians in society is to help people become acquainted with the underlying secret patterns. I have been working for several years on projects in crystallography, where we study crystal structures. But few people know that such periodic patterns come with severe constraints on their symmetry. In the plane, there are 17 different symmetry types, a fact well known even to the designers of the mesmerizing patterns of the Alhambra. Here you can find and experiment with such tilings in the plane, gaining insight into the intrinsic beauty of the so-called wallpaper groups, the crystal symmetries in dimension 2. The app interactively helps you design symmetric patterns with colors and shows how changes in the structure of the unit cell propagate via symmetry. nasqret.github.io/symm/ In dimension 3, if you look into International Tables for Crystallography, Vol. 1, you will find a theorem due to Schoenflies and Fedorov stating that there are 230 such symmetry types, a cornerstone of modern chemistry. Beyond that, in dimensions 4 and higher, a count can be made, but it requires a proof of the general theorem due to Frobenius and Bieberbach. This was an answer to the first part of Hilbert’s famous eighteenth problem. One of the fun consequences of such a classification is that in dimensions 2 and 3, 5-fold symmetry is forbidden in regular periodic arrangements. Intrinsically, this fact is related to the existence of matrices with a fifth-root-of-unity eigenvalue. For integral matrices, this is possible only in dimensions 4 and higher. If you generalize the square and cube tilings to dimensions 4 and 5, obtaining hypercubic tilings, the 5-fold symmetry pattern emerges. Skew projections of the 5D hypercubic tiling onto a 2-dimensional plane give rise to a quasicrystalline tiling known as the Penrose tiling. You can find such patterns in front of the Andrew Wiles Building at the Oxford Mathematical Institute. In later posts this summer, I will take a deep dive into group homology, a modern tool for studying the geometry of crystals. There are still many open questions, for example, how many symmetry types exist exactly in dimensions beyond 6. This is still largely unknown; at present, we only have asymptotic lower bounds.

THE SYMPOSIUM PUZZLE: The final dinner of the symposium was less a banquet than a convergence theorem that had failed to be uniform. Five luminaries -- Hardy, Poincaré, von Neumann, Gödel, and Ramanujan -- sat in a row at the head table, each in a different jacket, each with a different drink, each newly returned from a different lecture tour, and each guarding a different mathematical instrument as though it were a proof of the Riemann Hypothesis. Hardy sat brooding at the far left in herringbone, one hand curled around an espresso, the other resting upon an antique abacus whose beads he refused, on principle, to move. Immediately to his right sat a severe scholar in charcoal, upright as a metronome and no more companionable. Poincaré, ever the classicist, wore tweed. Farther down the line, Ramanujan (newly back from Göttingen) sat resplendent in navy, sipping tea and turning a golden compass over in his fingers as though it might draw identities straight out of the air. The navy jacket sat immediately to the left of the pinstripes, a juxtaposition that pleased no tailor present. The guest who had lectured at Cambridge, meanwhile, was the one in herringbone. When the conversation turned from foundations to apparatus, the scholar fresh from Princeton began boasting of a brass astrolabe he had recently acquired. Seated right next to him, the Göttingen speaker sneered that the workmanship was inferior to what one found on the Continent. Not to be outdone, von Neumann slapped an ivory slide rule onto the table with algorithmic enthusiasm. Gödel, with characteristic gravity, raised a glass of port in a toast that seemed prepared for its own incompleteness. The scholar just back from Oxford preferred brandy and, being full of it, soon leapt onto the table to make a point that no one had invited. In the ensuing disorder, a fellow guest's black coffee went flying. That black coffee, in the left-to-right order of cups along the table, had been sitting somewhere between Hardy's espresso and Ramanujan's tea. By morning the hall was deserted. Under the table lay four instruments: the antique abacus, the brass astrolabe, the ivory slide rule, and the golden compass. The silver caliper was gone. Who possessed each instrument -- and who had been carrying the missing silver caliper?

1/ new on arXiv: sheaf-theoretic clearing in financial networks... arxiv.org/abs/2605.15778 2/ eisenberg-noe (2001) defined a fixed-point equation for who pays whom when everyone owes everyone... a standard tool in systemic risk. the question: what is it, structurally? 3/ answer: it's a finite limit. liabilities form a sheaf on a directed hypergraph; the clearing vector is the global sections object. tarski, banach, kleene all recover as instances depending on the coefficient category... 4/ the headline theorem (clearing invariance) says functors preserving finite limits transport clearings correctly. so changing what you mean by "payment" (in Pos, DCPO, …) is structurally controlled... 5/ H⁰ is the clearing. what lives in H¹?

Today @NSF announced a major investment of $1.5B towards a new model for scientific research🚨 NSF X-Labs will fund independent teams of researchers, engineers, and entrepreneurs to pursue bold milestone-based scientific challenges. This is how we revitalize America's scientific engine for the 21st century outside of traditional institutions, conducting science in a way that actually reflects the modern R&D ecosystem ⬇️ nsf.gov/news/nsf-announces-1…

voronoi diagrams are how you divide space when you know where everything is school districts or cell tower coverage the kind of problem you solve with coordinates, a global view, and a computer. a CSHL group just showed the chinese money plant solves it in its leaves.

Very excited to share a new milestone in AI for Math: Aletheia, powered by Gemini Deep Think, was just used to autonomously solve a Kirby problem! “Kirby’s list” is a “compendium of the most important unsolved problems in topology, the study of deformable shapes” (Quanta magazine). 🧵

We are delighted to unveil our research blog Rabdology at rabdology.ai, where we chart the jagged math-frontier of AI reasoning. This is our first post in a weekly series. Read on, and if you enjoy it, please subscribe! (Link at bottom of blog's main page.) The Three-Cylinders Problem: When AI models choose Beauty over Truth rabdology.ai/three-cylinders We pose a problem that a good geometry student can solve in twenty minutes. We gave it to four of the world’s most advanced AI models and watched what happened. Three of them got it wrong — and the way they got it wrong tells you something different about the state of AI mathematical reasoning than the usual benchmarks.

my dad's stage iv cancer is in full remission; it was just 6 weeks ago i thought i was gonna lose him... god is good. (the staff at penn medicine are also very good...)

Happy Mother's Day! Here's a 13th c. image of the mother of our Lord clocking the devil 💕