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We all dodged a bullet

Loading... Why am I seeing this? You are seeing this because the administrator of this website has set up Anubis to protect the server against the scourge of AI companies aggressively scraping websites. This can and does cause downtime for the websites, which makes their resources inaccessible for everyone. Anubis is a compromise. Anubis uses a Proof-of-Work scheme in the vein of Hashcash, a proposed proof-of-work scheme for reducing email spam. The idea is that at individual scales the addit

We All Dodged a Bullet

Loading... Why am I seeing this? You are seeing this because the administrator of this website has set up Anubis to protect the server against the scourge of AI companies aggressively scraping websites. This can and does cause downtime for the websites, which makes their resources inaccessible for everyone. Anubis is a compromise. Anubis uses a Proof-of-Work scheme in the vein of Hashcash, a proposed proof-of-work scheme for reducing email spam. The idea is that at individual scales the addit

Anything can be a message queue if you use it wrongly enough (2023)

Loading... Why am I seeing this? You are seeing this because the administrator of this website has set up Anubis to protect the server against the scourge of AI companies aggressively scraping websites. This can and does cause downtime for the websites, which makes their resources inaccessible for everyone. Anubis is a compromise. Anubis uses a Proof-of-Work scheme in the vein of Hashcash, a proposed proof-of-work scheme for reducing email spam. The idea is that at individual scales the addit

'Ten Martini' Proof Uses Number Theory to Explain Quantum Fractals

But in some ways, the proof was a bit unsatisfying. Jitomirskaya and Avila had used a method that only applied to certain irrational values of alpha. By combining it with an intermediate proof that came before it, they could say the problem was solved. But this combined proof wasn’t elegant. It was a patchwork quilt, each square stitched out of distinct arguments. Moreover, the proofs only settled the conjecture as it was originally stated, which involved making simplifying assumptions about th

Who does your assistant serve?

Loading... Why am I seeing this? You are seeing this because the administrator of this website has set up Anubis to protect the server against the scourge of AI companies aggressively scraping websites. This can and does cause downtime for the websites, which makes their resources inaccessible for everyone. Anubis is a compromise. Anubis uses a Proof-of-Work scheme in the vein of Hashcash, a proposed proof-of-work scheme for reducing email spam. The idea is that at individual scales the addit

Ongoing Lean formalisation of the proof of Fermat's Last Theorem

Fermat's Last Theorem An ongoing multi-author open source project to formalise a proof of Fermat's Last Theorem in the Lean theorem prover. Information about the project The project is currently being led by Kevin Buzzard. Until September 2029 it is being funded by grant EP/Y022904/1, awarded by the EPSRC. The project is hosted at Imperial College London. Kevin would like to extend many many thanks to both of these institutions for their ongoing support of this nonstandard research. General

Age Verification Doesn't Need to Be a Privacy Footgun

“Won’t someone think of the poor children?” they say, clutching their pearls as they enact another stupid law that will harm the privacy of every adult on Earth and create Prior Restraint that inhibits the freedom of speech in liberal democracies. If you’re totally ignorant of how things work, the proposal of “verifying you’re an adult” before you access adult content sounds, superficially, like a reasonable thing to do. But it’s a patently stupid idea at every level. Age Verification Makes Th

The math is haunted

July 30, 2025 For the past few months, I’ve been writing a lot of Lean. Lean is a programming language, but it is mostly used by mathematicians. That is quite unusual! This is because Lean is designed to formalize mathematics. Lean lets mathematicians treat mathematics as code—break it into structures, theorems and proofs, import each other’s theorems, and put them on GitHub. The big idea is that eventually much of the humanity’s mathematical knowledge might be available as code—statically c

The Math Is Haunted

July 30, 2025 For the past few months, I’ve been writing a lot of Lean. Lean is a programming language, but it is mostly used by mathematicians. That is quite unusual! This is because Lean is designed to formalize mathematics. Lean lets mathematicians treat mathematics as code—break it into structures, theorems and proofs, import each other’s theorems, and put them on GitHub. The big idea is that eventually much of the humanity’s mathematical knowledge might be available as code—statically c

Evaluating publicly available LLMs on IMO 2025

Introduction Recent progress in the mathematical capabilities of LLMs have created a need for increasingly challenging benchmarks. With MathArena, we address this need by evaluating models on difficult and recent mathematical competitions, offering benchmarks that are both uncontaminated and interpretable. Among these competitions, the International Mathematical Olympiad (IMO) stands out as the most well-known and prestigious. As such, an evaluation of the IMO 2025, which took place just a few

Verified dynamic programming with Σ-types in Lean

1. Introduction If you’ve taken an algorithms class, you have likely seen dynamic programming, specifically a technique called memoization. Memoization works to optimize recursive algorithms by caching the solutions to subproblems in a table, and when a subproblem is encountered, it queries the table instead of recomputing the solution. This gives us an exponential performance boost. This blog post will show how to solve a dynamic programming problem using memoization in Lean, and verify its c

Verified Dynamic Programming with Σ-types in Lean

1. Introduction If you’ve taken an algorithms class, you have likely seen dynamic programming, specifically a technique called memoization. Memoization works to optimize recursive algorithms by caching the solutions to subproblems in a table, and when a subproblem is encountered, it queries the table instead of recomputing the solution. This gives us an exponential performance boost. This blog post will show how to solve a dynamic programming problem using memoization in Lean, and verify its c

Proofs Without Words

The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery. Nichomauss' Theorem: the sum of the first cubes can be written as the square of the sum of the first integers, a statement that can be written as . Here, we use the same re-arrangement as the first proof on this page (the sum of first odd integers is a square). Here's another re-arrangement to see this: This also suggests the following alternative proof:

Writing a Truth Oracle in Lisp

This post assumes some familiarity with typed functional programming, Lisp, and formal logic. Today we will attempt to write a truth oracle in Lisp. By "truth oracle," I mean a program that can determine whether arbitrary mathematical statements are true or false. This might sound impossible, due to first-order logic being undecidable, but let's try anyway. Before that, though, we need to go over some required concepts. Extracting information from proofs First, sometimes, we can extract info

Peano arithmetic is enough, because Peano arithmetic encodes computation

$\begingroup$ PA is enough, because PA can encode computation. This is longer than I expected, and was made longer still by some browser crashes. But I'd been idly thinking of writing these ideas up. I hadn't for these reasons. It is a lot of work. What I have to say is obvious to logicians, and they would consider the detour into programming to only be a distraction. Computer programmers who can appreciate the programming detour, are mostly not that interested in logic. But this question hi

Password-spraying attacks target 80,000 Microsoft Entra ID accounts

Hackers have been using the TeamFiltration pentesting framework to target more than 80,000 Microsoft Entra ID accounts at hundreds of organizations worldwide. The campaign started last December and has successfully hijacked multiple accounts, say researchers at cybersecurity company Proofpoint, who attribute the activity to a threat actor called UNK_SneakyStrike. According to the researchers, the peak of the campaign happened on January 8, when it targeted 16,500 accounts in a single day. Such