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Category Theory Illustrated – Orders
Given a set of objects, there can be numerous criteria, based on which to order them (depending on the objects themselves) — size, weight, age, alphabetical order etc.
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No country for mediocre mathematicians
White lies of AI from a struggling geometer
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How machines learned precision
How do you make an accurate machine without one to copy? Explore flat surfaces, screw-cutting lathes, and millionth-of-an-inch measurements through interactive 3D figures.
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A Theory of Deep Learning
Where noise hides when deep neural networks generalize.
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Simons Foundation launches $15M effort to help mathematicians navigate AI era
The Simons Foundation today announced a new set of initiatives that will help the mathematics community navigate the rapid rise of artificial intelligence in the field. Under a partnership…
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A look at the math that OpenAI claims computes faster Fourier transforms
The OpenAI algorithm for the FFT is less a breakthrough and more a malicious exploit.
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NP-overrated
If you learned about NP-hard problems in university, your takeaway was probably this:
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Mathematicians disagree on the essential structure of the complex numbers (2024)
I discuss several commonly held perspectives on the complex numbers and explore how their differences engage with several aspects of structuralism in the philosophy of mathematics.
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What TLA+ can and can't check
Let's chill just a little bit on the "TLA+ will save AI from itself" narrative
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Introduction to Formal Verification with Lean Part 1
A tutorial Formal Verification using Lean for cryptographic engineers. Implement and prove correctness of the One-Time pad following a cryptography book.
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Progress on provable inference for large language models
Sharing progress on a powerful type of cryptographic proof known as a zkSNARK, which is uniquely fit for provable inference.
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The Dark Night of Mathematics
What are we really doing?
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The Navier–Stokes Millennium Prize Problem
On the Navier–Stokes Millennium Prize Problem introduces an impressive result from OpenAI, who used an unreleased model to produce a resolution to the Navier–Stokes existence and smoothness problem, one of …
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The microstructure of wealth transfer in prediction markets
Slot machines on the Las Vegas Strip return about 93 cents on the dollar. This is widely considered some of the worst odds in gambling. Yet on Kalshi, a CFTC-regulated prediction market, traders have wagered vast sums on longshot contracts with historical returns as low as 43 cents on the dollar. Thousands of…
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We have proof automation now
I've long had a soft spot for dependently-typed languages like Coq Rocq and Lean. They offer the possibility of a type system capable of encoding and enforcing arbitrarily subtle invariants. The sort of thing that, in regular languages, ends up (at best) as a comment, and which quickly gets lost as the size of the…
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Faster than earcut: triangulating 3.7M map polygons in 489 ms
We published the triangulator of our map engine as earstrip, an ISC-licensed C++ and WebAssembly library. How it came to be: the ear clipper rebuilt one idea at a time and timed on 3.7 million polygons of real map data.
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RSA-896
A post about RSA-896.
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Avoiding Trigonometry (2013)
Tutorials and articles of Inigo Quilez on computer graphics, fractals, math, art, demoscene, shaders and more.
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A decades-old bug in Knuth's long division (TAOCP Vol II, Algorithm 4.3.1D)
A long division story
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Image Compression
Exploiting the quirks of human vision to make images smaller, without looking awful.
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Only 17% of all 64-bit Integers are products of two 32-bit integers
In software programming, the product between two integers is often computed to a fixed number of bits with overflow. Consider 8-bit integers. If you multiply 127 by 127, you get back the number 1 as an 8-bit unsigned integer, with an overflow. The actual full product is 16129. To represent 16129, you typically use 16…
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The Q, K, V Matrices
At the core of the attention mechanism in LLMs are three matrices: Query, Key, and Value. These matrices are how transformers actually pay attention to different parts of the input. In this write-up, we will go through the construction of these matrices from the ground up.
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Knuth reward check
Twenty years ago I got a reward check from Donald Knuth.
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An interactive intro to quadtrees
An interactive exploration of quadtrees. Start with brute-force spatial search, build recursive space partitioning step by step, then use it for point lookups, range queries, nearest-neighbor search, collision detection, and image compression.
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Upper stage impacting the moon on 2026 August 5
On 2026 August 5, within a few minutes of 06:35 UTC, an upper stage (section) of a rocket used for a lunar mission will hit the moon. This may be of some (probably minor) scientific interest, and we may learn some things from it. It doesn't present any danger to anyone, though it does highlight a certain carelessness…
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Mr. Baby Paint and accidentally discovering a new cellular automata
On building Mr. Baby Paint, a radically simple co-op drawing app for toddlers & accidentally discovering a flood-fill cellular automata, plus developing pixel-fattening algorithms for bitmap fonts.
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It's OK to compare floating-points for equality
NB: The title of this post is an intentional clickbait. Even though I do stand for its statement, a more honest one would be something like: It's NOT OK to compare floating-points using epsilons.
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OpenAI, the Partition Principle, and Mathematics
By the time I got out of bed yesterday, I had three people ask me if I'd seen the OpenAI announcement that the Partition Principle does not (...)
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What mathematicians should know about the Lean Theorem Prover: reliability & AI
[This is a guest post by Thomas Hales. This blog post was initially written in a different file format and converted using AI. — T.] Mathematicians have been weighing in on what they value ab…
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Why are we even doing all this?
[This is a guest post by Rikab Gambhir. This blog post was initially written in a different file format and converted using AI. — T.] First, they came for the artists, and I did not speak out…