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Why I didn’t sign the Fields medallists’ letter
[This post has been cross-posted to Terence Tao’s blog.] When I was around 11 I heard for the first time about Fermat’s Last Theorem. I was immediately captivated by the problem stateme…
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Principia Mathematica is modern and insightful
Notes on Chapter 1 of Principia Mathematica
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Backtrack-Free Cursive
Making English more enjoyable to write.
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A beginning for mathematics
A positive vision for the future of mathematics: how our institutions can deepen human understanding as AI transforms mathematical research.
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I vibed a proof of Conway's conjecture
You can just prove things, apparently.
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A faster way to calculate the day of the week
A range of fast modulus techniques that beat compiler output
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Hilariously fast volume computation with the divergence theorem (2018)
The following presents a fast algorithm for volume computation of a simple, closed, triangulated 3D mesh. This assumption is a consequence of the divergence theorem. Further extensions may generalise to other meshes as well, although that is presently out of scope.
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Quantum computing bombshells that are not April Fools
For those of you who haven’t seen, there were actually two “bombshell” QC announcements this week. One, from Caltech, including friend-of-the-blog John Preskill, showed how to do quantum faul…
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What sort of maths are LLMs good at?
For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten majo…
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Finding all regex matches has always been O(n²)
every regex engine that promises linear time breaks that promise the moment you ask for all matches. the problem has been there since the 70s, hiding in the iteration loop.
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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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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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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.