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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…
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Will it be a golden age or a dark age for mathematics (and science)?
[This is a guest post by Francesco Maggi. This blog post was initially written in a different file format and converted using AI. — T.] Large language models (LLMs) and the research systems b…
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Introduction to spherical harmonics for graphics programmers
On your computer graphics journey, you will eventually run into some paper or code mentioning spherical harmonic functions. They are indeed a very useful tool: with just a few coefficients, they allow us to approximate a given function defined on a sphere. This can be very useful for modeling complex lighting.
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PageRank explained
Picture this, the year is 1996. You find yourself frustrated with the incumbent search engines like AltaVista, which primarily does a content-based search (it'll give you an article on "Hotels for Chickens" if you search "Hotels" because the word matches). There's gotta be a better way, right? Well, in hindsight, of…
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Faster Than Dijkstra?
Does a new algorithm that improves on Dijkstra'a classic one actually deliver a practical benefit for networking?
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The quadratic sandwich
If you have ever tried to minimize a function with gradient descent, you probably noticed that some functions are a joy to optimize and others are a nightmare. The difference often boils down to two properties: strong convexity and L-smoothness. These two concepts define a “sandwich” of quadratic bounds around your…
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Not all elementary functions can be expressed with exp-minus-log
a personal blog
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The Entropy of a Markov Chain
A more explicit example
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The simple geometry behind any road
How does the geometry of a road unfold?
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Saving Jet Fuel
Benchmarks & Tips for Big Data, Hadoop, AWS, Google Cloud, PostgreSQL, Spark, Python & More...
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Why Mathematica does not simplify sinh(arccosh(x))
When Mathematica seems to not to know how to simplify an expression, it may be doing the right thing, being more sophisticated than you expect.
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Starting from scratch: Training a 30M Topological Transformer
Tauformer is a topological transformer (see paper ) that replaces dot‑product attention with a Laplacian-derived scalar (taumode) per token/head, then attends using distances in that scalar space. Below is a post-style overview of the idea and the first training signals from a 30M-parameter run.
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Two-Stack Sliding-Window Aggregation
An aggregation is some kind of summary of a set of data. This can be the sum, length, minimum, etc. It is quite common to want to calculate such a summary repeatedly, e.g. “the maximum noise level in dB for the past 30 seconds” for a nuisance detector. In such a case we say there is a sliding window over our data, and…
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An interactive intro to Elliptic Curve Cryptography
A hands-on introduction to elliptic curve cryptography. Start with curve geometry, build point addition and scalar multiplication, see why ECDLP is hard, and then use that math in ECDH, ECDSA, and ECIES.
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Differential Heuristics
In 2005 Google showed off Google Maps, which lets you drag the map around instead of reloading the page like MapQuest and others used. That’s the feature that got everyone’s attention. But what got my attention was a feature they added in 2007 where you could drag the start/end points on a route [1] , and it would…
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Can guitar frets perform multiplication?
I’m sure that some pictures are worth a thousand words, but others trigger a whole lot of puzzlement. Such was the case with the cover of a book I recently bought entitled Calculating with Tones: The Logarithmic Logic of Music:
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The adder at the heart of Intel's 8087 floating-point chip
In 1980, Intel released the Intel 8087 floating-point coprocessor, a chip that could make math up to 100 times faster. As well as arithmetic...
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How to store a chess position in 26 bytes (2022)
Made in Notion, the connected workspace where AI agents and teams organize docs, projects and knowledge together.
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The case against geometric algebra (2024)
Every once in a while the internet gets talking about Geometric Algebra (henceforth GA) and how it’s a new theory of math that fixes everything that’s wrong with linear algebra and multivariable calculus. When I come across this stance I am compelled to respond with something like: “wait wait, it’s not true! GA is…
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Using calculus to do number theory
Calculus is the mathematics of approximations of continuous quantities ; number theory asks exact problems about discrete quantities. These seem, therefore, to be unrelated studies.
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Continuous Diffusion Language Models (CDLM's)
Fully discrete methods dominated for a few years, but language models based on continuous diffusion are making a comeback.
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Is mathematics about to enter the conservatory?
Math as cultural institution.
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Softmax, can you derive the Jacobian? And should you care?
Practice machine learning engineering with hands-on coding challenges. Implement neural network components, backprop, and more — from scratch.
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What does the Riemann zeta function have to do with the distribution of primes?
One of the oldest theorems in mathematics is that there are infinitely many primes. Euclid (the geometer!) proved this around 300 BCE, and for centuries afterwards the story ended there. There are infinitely many primes -- so what else is there to ask about them?
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Topologist's Map of the World
Reddit user xilefakamot created this in 2020 — a “topologist’s map of the world.” It depicts international borders and nothing else — the shapes, sizes, and distances between countries are forsaken in order to present the pattern of their borders as efficiently as possible. “No calculations per se – I sketched out a…
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A Visual Intro to Complex Numbers
A visual and intuitive intro to complex number system providing a historical overview of how it was developed and why it is useful.
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Anatomy of a Lean proof for software engineers
Intro I recently worked through a problem from a theory of computation textbook that asked me to prove a property of a language using finite automata. The informal proof is a simple constructive proof where you build an automaton and show that it recognizes the language. This is kind of similar to program…
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Unicode's transliteration rules are Turing-complete
Claim: as specified, the UTS #35 transliteration-rule language is Turing-complete. This is an exception in Unicode, where algorithms are deliberately bounded and always terminate. The specification defines a revisiting cursor with no rewrite limit. ICU's implementation implements a per-call cap, but the spec itself…
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Learning Lean: Part 1
Motivation I’ve been captivated by the recent movement to popularize mathematics formalization through the Lean theorem prover, and this year I’m diving deeper into learning it. For those unfamiliar with this revolution, I highly recommend watching Kevin Buzzard’s talks on YouTube for an overview of why formal…