Normal Science

247A, Notes 1: Rearrangement-invariant spaces

Disclaimer: due to current events, I have not been able to devote as much time to lecture notes preparation as I would have liked, so I apologize in advance for the unpolished nature of the text below, which has been largely recycled from previous lecture notes I have written. This is the first set of lecture notes for my graduate course 247A, “Fourier analysis”. The course name is rather general, but I will focus the course not on the Fourier transform per se, but on the closely related topic o...

Randomized query complexity can beat certificate complexity

Abstract page for arXiv paper 2609.15063: Randomized query complexity can beat certificate complexity

SAIR’s Open Math Model initiative

Last year I co-founded the Foundation for Science and AI Research (SAIR) with some private donors to create a non-profit organization that could support responsible uses of AI in mathematics and the other sciences, independent of the major AI companies. Initially, our resources were rather limited, and our main activities consisted primarily of podcasts, short events, and competitions. However, we have been slowly trying to expand in scale, ambition, and logistical capacity; for instance, we wer...

Mathematicians Build Long-Awaited Graph Sandwich

In 2004, two mathematicians hypothesized a powerful kind of sandwich. They were studying graphs, which are collections of points (called vertices) and lines (called edges). Graphs might represent anything from social groups to the internet to neurons in the brain. The mathematicians hoped to understand properties of one type of graph — a type that’s ubiquitous in mathematics and computer science… Source

Becoming a benchmark

[This is a guest post by Talia Ringer. This blog post was initially written in a different file format and converted using AI. — T.] When I was wrapping up graduate school in computer science in Spring 2021, I was given access to a curious little programming model on OpenAI’s “playground.” The model, called Codex, took in natural language text and generated programs from that text. The task of automatically generating programs given an expression of programmer intent had been one of many major q...

Proofs, Prompts and Posts

[This is a guest post by the editors of the communal blog Proofs and Prompts. This blog post was initially written in a different file format and converted using AI. — T.] In the mathematical community, AI became a central conversation topic months before we started our communal blog, called Proofs and Prompts. Some of us were already experimenting with it, whereas some of us were skeptically watching from the sidelines. Inevitably, we all had our moment of realisation: be it Navier–Stokes, the ...

Fast math/slow math

[This is a guest post by Ben Antieau, crossposted from his blog. This blog post was initially written in a different file format and converted using AI. — T.] To resolve a central tension in the practice of mathematics by academics, I suggest the parallel with fast food and its attendant slow food movement. Three hearts Mathematical work comprises many activities and experiences. Among my favorites are: Reading a single paper over the course of several months. The thrill of the chase, when the s...

The technical debt of AI-generated mathematics

[This is a guest post by Henry Cohn. This blog post was initially written in a different file format and converted using AI. — T.] In this essay I’d like to discuss the role of AI-generated solutions in mathematics and why many mathematicians are justifiably concerned about a mismatch of values. Let’s start with the groundwork, so we’re all on the same page regarding what mathematics is and what is needed for progress. The role of understanding Academic mathematics is about understanding, not ju...

Why Alberta doesn’t have rats

This piece was featured in Issue 24 of Works in Progress. Subscribe today to get Issue 26, and an issue every two months after, delivered straight to your home or workplace.Most of us accept rats as a fact of life. They live in tunnels and sewers, gnaw through walls, contaminate food, and resist nearly every attempt to push them back. A 2023 estimate put New York City’s rat population at about three million animals, roughly one rat for every three human residents. Almost nowhere do authorities t...

A Misalignment of AI in Mathematics

I am proud to be among the list of 25 initial signatories — all Fields Medallists — to the declaration below, which grew out of discussions between ourselves over the last week. We have…

US–Indian space mission maps extreme subsidence in Mexico City

Why are neural networks and cryptographic ciphers so similar? (2025)

Several neural network design principles are almost identical to cipher design principles.

Fun with polynomials and linear algebra; or, slight abstract nonsense

Longest Path Search About Code Fun Fun with polynomials and linear algebra; or, slight abstract nonsense Posted 2026-04-30 This is mostly a bunch of notes to myself (with some slight expansion) and is a combination/extension/simplification of theorems/ideas/constructions from a bunch of texts, including Wistbauer’s “Foundations of Module and Ring Theory” and Fuhrmann’s “A Polynomial Approach to Linear Algebra”, along with others that at this point I don’t recall. While most of the things here ar

The gauge broke: devs felt 20% faster with AI, measured 19% slower

For two years I argued the feeling of AI speed had come apart from the fact of it, from watching my own teams. This summer it stopped being an anecdote. A controlled trial measured experienced developers feeling about 20% faster while running about 19% slower. The instrument we steer by reads backward.

The Mathematical Dance Inside Plant Cells

The sunlight-collecting organelles known as chloroplasts solve a packing problem: how to optimize photosynthesis without sustaining damage from dangerously intense rays.

Easy Random Trees

Rigorous Nonsense Home Archive Easy Random Trees Posted on February 27, 2026 by Brandon Wilson Tags: math, apl, trees Can you think of a way to efficiently generate a random plane tree? Richard P. Stanley in his book Catalan Numbers has a really nifty combinatorial proof of why Catalan numbers have the formula \[ C_n = {1 \over n+1}{2n \choose n} \] The standard proof uses generating functions applied to an inductive definition of the Catalan numbers, which frankly does little to illumiate their

OpenAI's claimed disproof of Connes' Rigidity Conjecture is invalid [pdf]

The extended predicative Mahlo universe in Martin-Löf type theory (2023)

Finite time blowup with smooth forcing term for the incompressible porous medium

There’s some exciting very recent work by Alpöge and Buckmaster, building upon prior work by Córdoba and Martínez-Zoroa, in the general topic around the infamous global regularity problem for…

Every shape can be cut exactly in half with one straight line

A simple consequence of continuity, a surprising connection to topology, and the gap between knowing a solution exists and actually finding it.

Why Are Rivers So Mathematical?

A simple scaling law brings order to the chaos of flowing water, rock, and sediment. New findings have extended the law even further.

How Is Compression Prediction?

When does lossless compression reduce to prediction, and what must already be fixed before that equivalence applies?

Natural history on canvas: Brueghel knew about bird-eating noctule bats

Things I Won't Work With: Straight Dimethyl Zinc

A 13th-Century Enumeration Algorithm, Ignored for 700 Years

In Or ha-Sekhel, the Kabbalist Abraham Aboulafia prescribes a systematic way to run through every arrangement of the letters of a word. With only two or three simple rules, he describes what we would now call a recursive algorithm for generating permutations.

Deconstructing Datalog

home datafun talon reviews résumé rss Deconstructing Datalog In September 2022, after two rounds of revisions, I submitted the final version of my PhD dissertation, Deconstructing Datalog. Datalog is a logic programming language from the ’80s that augments relational algebra with recursive queries. It has both simple semantics and efficient implementation strategies. Like Lisp and the Velvet Underground, its influence exceeds its popularity; its ideas are still being absorbed into the mainstream

Classical chess ranks 562nd of 960 starting positions after 460,800 games

Round 480 closed exactly half the championship. What half a million games showed — and why the classical setup came out 562nd of 960.

The Riemann Hypothesis – interactive explanation

An interactive journey from 'what is a prime number?' to a genuine understanding of the most important unsolved problem in mathematics.

Topological Defect Propagation to Classify Knitted Fabrics

“Deep theorems were scarce and difficult and so became an effective mechanism to identify deep thought. AI has broken this system.”

[This is a guest post by Bryna Kra. This blog post was initially written in a different file format and converted using AI. — T.] For generations, mathematicians have treated the production of theorems as a clear measure of success. The stronger the theorem, the deeper the proof, the more surprising the connections, the greater the achievement. Positions and prizes are based on these theorems and the mathematicians making the breakthroughs set the directions for future research. But theorem prod...

The status of the Hodge conjecture

[This is a guest post by Claire Voisin. This blog post was initially written in a different file format and converted using AI. — T.] Hodge classes can be defined on any compact complex manifold . They are rational Betti cohomology classes on of even degree (eg combinations with -coefficients of classes of oriented codimension closed submanifolds of ) which satisfy a delicate “Hodge condition” necessary for them to be combinations with -coefficients of classes of complex submanifolds (and more g...

Principia Artificialis – Mathematical Foundations of Artificial Thought

An open research program exploring the mathematics of artificial thought: information geometry, topology, dynamical systems, and thermodynamics applied to AI inference. - holland202/Principia-Artificialis

On the existence of non-sofic groups

[This is a guest post by Andreas Thom. This blog post was initially written in a different file format and converted using AI. — T.] When I woke up on August 1st, 2026, I had received a few emails from colleagues asking for my opinion on a remarkable result that had circulated the previous day. The result was a solution to a long-standing open problem in geometric group theory, specifically the existence of a non-sofic group. I was astonished and at the same time, looking at the first draft, als...

A Severe Misalignment of AI in Mathematics

I am proud to be among the list of 25 initial signatories — all Fields Medallists — to the declaration below, which grew out of discussions between ourselves over the last week. We have also posted our declaration on this web page, and (similarly to the Leiden declaration) invite further signatures. (It is unfortunate that we did not have the time to have a more consultative process, as with Leiden; but we decided that the urgency of the situation was such that we needed to release a statement s...

On the Hodge conjecture

[This is a guest post by Burt Totaro. This blog post was initially written in a different file format and converted using AI. — T.] These are strange times for mathematicians. It now seems possible that AI companies will burn through vast resources in order to prove some new fact about the Hodge conjecture. I’d like to discuss the current status of the Hodge conjecture, in order to think about the value to the mathematical community of exploring such hard problems. Thanks to Terry Tao for sugges...

Why Do These Fossil Shells Flip Their Spirals Every Few Millennia?

For thousands and sometimes millions of years, marine plankton all over the world built their spiral shells in one direction. Then, suddenly, the spirals switched direction at the same time, everywhere, only to switch back again later. These microorganisms are types of foraminifera, or forams. Found in all oceans, from the tropics to high latitudes, they’re among the most abundant eukaryotic… Source

SAIR competition: Andrews-Curtis challenge

[This is a guest post by Lucas Fagan. This blog post was initially written in a different file format and converted using AI. — T.] I am excited to announce the Andrews–Curtis Conjecture Challenge, which opens today. This challenge is a collaboration between the SAIR Foundation and the Math-AI group at Caltech, organized by Sergei Gukov, Terence Tao, and myself. The Andrews–Curtis conjecture is one of the most prominent open problems in combinatorial group theory and also has deep connections to...

Where Has Construction Automation Been Successful?

Construction is famously labor intensive: direct labor makes up close to 50% of the cost of constructing a new single-family home in the US, compared to around 6 to 8% of the cost of manufacturing a car. Unsurprisingly, there’s long been interest in automating the construction process to reduce the amount of labor required, and therefore the cost of housing. The earliest attempt at automating construction I’m aware of dates to 1853, with a plastering machine invented by Isaac Hussey, and since t...

Stable singularity of the Euler equations on R^3

[This is a guest post by Anima Anandkumar. This blog post was initially written in a different file format and converted using AI. — T.] The last few days have been a flurry of activity. After months of sleepless nights and diving deep into different mathematical and computational tools to tackle the open problem of singularity in fluid dynamics, we were finally ready to send our work on Euler equations (without forcing) to Tom Hou and Terry Tao for feedback. Almost immediately I heard back from...

The Four-Color Theorem Gets a Rare New Proof

Some math problems continue to haunt researchers long after they’ve been solved. A proof emerges, is even celebrated, and yet dissatisfaction lingers. Perhaps the argument is too convoluted — the hunt persists for the elusive one-page paper — or perhaps it fails to give a deeper theoretical insight into why something is true. Whatever the reason, mathematicians return, again and again… Source

Crowdsourcing a list of general resources on AI and mathematics

Given current events, I think it is worthwhile to start collecting useful online resources with regards to AI and mathematics in general. I will start a list below, but ask contributors to contribute further links in the comments. Note that while I personally do not agree with 100% of the content of every single link below, I find each of them interesting enough to be worth mentioning here. I am excluding my own writings on these topics, which can be found here (and summarized here) instead. Res...

Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations

There’s some exciting very recent work by Alpöge and Buckmaster, building upon prior work by Córdoba and Martínez-Zoroa, in the general topic around the infamous global regularity problem for the incompressible three-dimensional Navier-Stokes equations. It is now widely expected that it should be possible to construct smooth initial data and smooth forcing term that would make these equations develop singularities in finite time; and it should even be possible to do without the forcing term. Whi...

How Will the 21st Century ROAD to Housing Act Affect Housing Supply? Part III

Via Wikipedia.This is my third and final essay looking at the 21st Century ROAD to Housing Act, and how it’s likely to affect housing supply. As I’ve noted in my previous two essays, this is a sprawling piece of legislation, with 59 sections spread across 12 different “titles.” So far we’ve looked at Titles I, II, and III, which together contain 24 individual sections. In this post, I’ll look at the remaining nine titles of the package. There are a lot of sections (35!), but many of them don’t a...

Live from ICM 2026: What Is Math For in the Age of AI?

Mathematicians are witnessing a profound shift in their field. AI systems are now producing proofs, spotting connections between distant fields, and, in a few cases, solving problems that had stumped mathematicians for decades. The pace of progress over the past several months has raised challenging questions: What can AI systems actually do? And what happens to the more human… Source

In an Age of AI, a Physicist Seeks What Endures

In July 2026, the U.S. Department of Energy selected 278 projects to fund as part of its Genesis Mission, meant to incorporate artificial intelligence into the scientific process. One project will use AI to help physicists search for a vanishingly rare particle transformation — a muon becoming an electron — by controlling different aspects of the Mu2e experiment at Fermi National Accelerator… Source

Genome Duplication Is a Radical Evolutionary Gamble

No larger than the head of a match, Potamopyrgus antipodarum doesn’t look very impressive. Few people visiting New Zealand’s Lake Alexandrina even notice the tiny gastropods littering the shores. The snail’s diminutive size, however, conceals a massive secret. Sometime in the recent past, its genome doubled. Instead of having two sets of chromosomes and two copies of every gene… Source

Reading List 08/29/26

Morning in the Tropics by Frederic Edwin Church, via WikiArt.Welcome to the reading list, a weekly roundup of news and links related to buildings, infrastructure, and industrial technology. This week we look at sinking London homes, US magnesium manufacturing, the World Humanoid Robot games, catastrophic flooding in Nepal, and more. Roughly 2/3rds of the reading list is paywalled, so for full access become a paid subscriber.Housing and CitiesNew home sales are down 6.3% year over year. [Calculat...

Does Computer Science Need Computers?

The pioneering computer scientist Edsger Dijkstra, winner of the 1972 A.M. Turing Award and inventor of one of the most iconic algorithms in all of computing, was nothing if not opinionated. Certain programming languages drew his ire, for example: He once dubbed Fortran “the infantile disorder” and stated that “the use of COBOL cripples the mind; its teaching should, therefore, be regarded as a… Source

How the Strategic Petroleum Reserve Works

The US Strategic Petroleum Reserve (SPR) is the largest crude oil storage facility in the world. It’s capable of storing 714 million barrels of oil, slightly less than 10% of the US’s annual oil consumption. Built in response to OAPEC’s 1973 oil embargo, the SPR was created to insulate the US from oil supply shocks: to make up for shortfalls and prevent steep price increases should oil imports be interrupted. In the years since its creation, it has become one of the most frequently used energy p...

Rotating needles in space: the road to the Kakeya conjecture, and why it matters

A recent tradition of the ICM is to have a non-technical popular article written for each of its medallists. (This is separate from the older tradition of having a laudatio for the winner, which is similar but aimed at a more advanced audience.) I was approached to write such an article for Hong Wang on the occasion of her Fields Medal. I have uploaded my initial version of this article, “Rotating needles in space: the road to the Kakeya conjecture, and why it matters“, to the arXiv. Chris Sogge...