This Week
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
Older
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
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.
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.
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
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.
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
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
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...
A future of programmable medicine
This piece is featured in Issue 25 of Works in Progress. Subscribe today to get an issue every two months delivered straight to your door.Jacob Witten explains why siRNA will change medicine as we know it.If you want a picture of biomedical research, imagine a boot stamping on a biologist’s face – for decades. The boot is labeled ‘medicinal chemistry’. This should not be taken as an insult to medicinal chemists, who heroically navigate maybe the hardest field in biomedicine.1 Rather, what I mean...
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...
Mathematical Discourse
Mathematical Discourse is a new online, peer-reviewed mathematics journal, publishing videos of mathematics research talks of the highest quality. Our inaugural scientific and editorial boards are listed at the end of this message. Our goal is to promote and nourish a culture which values communication as an essential part of the research process. Giving a mathematical talk captures the human side of mathematics in a unique way, and supporting these human aspects of our discipline will be ever m...
‘Huge Breakthrough’ in the Math of Imbalance
One does not need a doctorate in mathematics to split 12 eager trivia buffs into two competitive teams. But consider that each person arrives with unique strengths and liabilities: One may be a geography obsessive with no ear for music, another could be a naturalist who doesn’t own a television, and another could be a cinephile who never reads. Balancing traits between two camps becomes a lot… Source
Palomar – a registry of Lean verified mathematics
In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking that a given Lean repository actually proves the claimed statement is somewhat non-trivial, especially for an audience which is not expert in the use of Lean: one has to first check that the claimed formal Lean statements have proofs that typecheck, that the proofs do not contain any “cheats” such as addi...
Theory of Fluids Enters the 21st Century
In the second half of the 20th century, a conceptual tsunami swept through physics. The discovery that our world emerges from a microscopic world of molecules, which emerges from an even more microscopic world of subatomic particles (which in turn emerges from even stranger stuff) triggered the rewriting of our theories of matter. But the revolution didn’t reach fluids. Source
How Will the 21st Century ROAD to Housing Act Affect Housing Supply? Part II
This is the second installment of my look under the hood of the recently passed 21st Century ROAD to Housing Act, looking at what each provision actually does and trying to suss out its actual impact on housing supply. You can read my previous essay on Title I of the Act here. This time I’ll look at Title II, “Building More in America,” and Title III, “Manufactured Housing For America,” which have 13 and 4 individual provisions respectively. Some of the most consequential chunks of this legislat...
A digestion of the proof of Sendov’s conjecture
This post concerns the following conjecture of Sendov, as well as its strengthening by Phelps–Rodriguez: Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with . Conjecture 2 (Phelps–Rodriguez conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with , unless is on the unit circle and is a scalar ...
Graduate Student Proves a Quantum Uncertainty Principle for Fractals
At the quantum scale, tiny particles behave in bizarre ways. One reason for this is the uncertainty principle, which says that the more you know about where a quantum particle is, the less you can know about how fast it’s moving, and vice versa. Recently, this rule got a rare upgrade. The new uncertainty principle relates to fractals, shapes that remain equally complex no matter how much you… Source
Why Are Rivers So Mathematical?
A river has my heart. It’s not the austere, black Thames winding through London, where I was born, but a lazy green one 5,000 miles away, where I spent my adolescence: the Blanco River in Texas. My maternal ancestors have dipped into its waters for generations, as I have on countless summer days. The Blanco is a tributary of the San Marcos, which flows into the Guadalupe, and on into the Gulf of… Source
Neutrinos From Deep Inside Earth Provide a New Picture of the Mantle
In a laboratory 2 kilometers underground, a crane lowers Matt Depatie, a detector technologist, through a hatch into a white-walled cavern filled with about 7,000 tons of ultrapure water that glows as blue as wiper fluid in the light. “Splashdown,” Depatie says over a radio as he steps into an inflatable raft waiting below. Normally, the cavern is one of the darkest places on Earth, but today… Source
What Will the 21st Century ROAD to Housing Act Do for Housing Supply?
Home under construction, via Wikipedia.A few weeks ago the 21st Century ROAD to Housing Act became law. This is an enormous housing bill that’s been in the works for roughly a year — I first discussed it on Statecraft back in 2025 — and folks are optimistic that it will, per its description, “increase the supply of housing in America.”But actually understanding what, specifically, the law will do is tricky. Like many large pieces of legislation, this is an amalgamation of many smaller laws: the ...
The upper limits of biology
This piece is featured in Issue 25 of Works in Progress. Subscribe today to get Issue 25, and an issue every two months after, delivered straight to your door. 9th June 1973: Secretariat leaves the field behind at Belmont Park. Image credit: Getty Images.In June 1973, chestnut colt Secretariat ran the Belmont Stakes so fast that no horse has matched him since. In less than two and a half minutes, Secretariat flew along the track, finally pulling away to win by 31 lengths, nearly three quarters o...
A partial digestion of the HRT counterexample
A function of one variable can be translated in space by a spatial shift to obtain a new function and also modulated in frequency by a frequency shift to obtain a new function One can compose these two operations to obtain a time-frequency shift: As per the time-frequency uncertainty principle, the two shifts do not quite commute with each other. For instance, we have As such, is not a representation of the abelian group , but rather a portion of the Weyl representation of the Heisenberg group, ...
Corals Spin Tiny Vortices to Get Oxygen, but Not if It’s Too Hot
At first glance, corals present as little more than colorful rocks — piles of lobes, stalagmites, and branches poking out from the seafloor. They are anything but. Corals are complex creatures that form enduring colonies, and just like other animals they need oxygen to live. Across the living surface of coral, a frantic dance of survival takes place, invisible to our eyes and unknown to science… Source
A Stupid Idea for AI Alignment We Came up with by Looking at the List of Specification Gaming Behaviours
Creatures bred for speed grow really tall and generate high velocities by falling over. An evolved player makes invalid moves far away in the board, causing opponent players to run out of memory and crash. A game-playing agent accrues points by falsely inserting its name as the author of high-value items. These bizarre exploits and dozens more can be found in the list of specification gaming behaviours [sic; British], a document put together by DeepMind Safety Research. “A reinforcement learning...
Why the Legendary Erdős Problems Are Falling to AI
On May 20, 2026, OpenAI made an announcement that shook the mathematical world. An internal AI model — one not available to the public — had come up with a counterexample to the “unit distance” problem, a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician. Erdős posed thousands of questions, but this one was special: It was both simple to explain and… Source
Indirect lessons from human alignment
(Inspired by a post from Eli Tyre.) Many people make some variant of the following argument: Evolution is an “outer optimizer”. It is trying to make us maximize reproductive fitness. We are “inner optimizers”. We just do what feels good. But what feels good has been set by evolution, which is hoping that it will make us maximize reproductive fitness. But we don’t maximize reproductive fitness. In fact, birth rates are dropping everywhere. Therefore evolution failed. The standard interpretation i...
Is AI Reasoning Right for the Wrong Reasons?
I’ll just say it: What the hell is going on with AI “reasoning”? Sorry for the air quotes. That punctuational side-eye was more common in 2024, when the specially trained cousins of LLMs now known as “large reasoning models,” or LRMs, were still new. Nowadays it may seem downright churlish, though, given that a “general-purpose reasoning model” from OpenAI solved a famous open mathematical… Source
Why Is Everyone Trying to Build a Solid-State Battery?
A battery technology that’s getting a lot of attention is solid-state batteries, lithium-ion batteries that replace the liquid electrolyte with a solid material. Chinese battery manufacturer CATL alone had more than 1,000 people devoted to solid-state battery research as of 2024, and battery manufacturers like BYD, LG, and Samsung are also working on the technology. US and European startups making solid-state batteries have collectively raised over $4 billion as of 2025.Solid-state batteries hav...
So you want to use plants to reduce CO₂
Humans make carbon dioxide. Carbon dioxide is bad for cognition. But plants turn carbon dioxide back into oxygen. And plants are the one true home decoration strategy. So maybe if you get a lot of plants, you can you can keep carbon dioxide in check and keep your brain working? It’s theoretically possible. It’s probably just barely possible in practice. But it won’t be easy. People produce ~1 kilogram of carbon dioxide per day. That’s around 5.7 × 10²³ molecules or 0.948 moles per hour. (You may...
Can We Lower Construction Costs with Cheaper Labor or Materials?
3D printed structure, via Wikipedia.Last time in our series on the nature of the construction productivity problem, we looked at economies of scale, finding that capturing economies of scale in homebuilding is difficult. Economies of scale primarily operate on the difference between the costs of the material inputs to some process and the cost of the final product. With housing construction, this ratio is already pretty low, not much higher than what we see in high-volume manufacturing industrie...
A digestion of the Jacobian conjecture counterexample
The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse). The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be deri...
Does creatine make you smarter?
Is creatine a weird steroid-like hormone or drug? No. Creatine is a nutrient. Most omnivores eat a gram or two per day from meat. Your body also synthesizes a gram or two per day. You need creatine to deliver energy inside of cells. It is normal and non-weird. Does creatine increase testosterone? Unlikely. This concern comes from one study in 2009 on 16 male rugby players.1 But that study is considered extremely suspect. There have been at least twelve other studies that all found no change or p...
How Predictable Are Laws?
An enormous amount of ink has been spilled (some of it by me) on the trials and tribulations of complying with the National Environmental Policy Act, better known as NEPA. NEPA is what requires projects to perform years-long, thousand-plus page environmental impact studies before construction can begin, and suing a project for an insufficiently detailed environmental study is one of the chief ways environmental groups are able to slow down or stop projects they don’t like. And NEPA’s influence g...