Mathematics

Naming the Galois group of a degree-24 integer polynomial ordinarily demands resolvent computations inside a computer algebra system. This paper presents a statistical alternative that needs only polynomial factorisation modulo small primes, and evaluates it against 576,682 polynomials whose groups the SAIR IGP24 evaluation server computed independently in Magma. The method reads the multiset of Frobenius cycle types at 60 primes as a sample from the group’s own cycle-type distribution, which the Chebotarev density theorem licenses, then scores that sample against empirical profiles of candidate groups drawn by product-replacement sampling. Within its domain the classifier reaches 69.0% top-1 and 88.7% top-3 accuracy at 53 ms per polynomial.

A competition organised by Terence Tao and Damek Davis asked whether mathematical reasoning can be compressed into a page a language model reads before answering. Then its second stage removed the thing that made the first stage easy to fake: an answer was no longer worth anything unless it arrived with a proof a machine would check. This is what I built for both stages, and the moment that convinced me a confident answer and a correct one are completely different products.

There is a question in mathematics that has been open since the 1800s, and for one slice of it you can now attack it by search. I built a factory that produced degree-24 polynomials, submitted ten thousand scoreable pairs to a competition organised with the LMFDB and Terence Tao, and finished 54th of 256 with a score of 2.36. The interesting part is not the rank. It is that I can prove, with numbers, exactly why the score was small, and the answer says something about how these searches should be run.

A calculator multiplies two thousand-digit numbers and takes the remainder without thinking about it. Ask a neural network to do the same thing and it becomes an open research question, one that Terence Tao helped set as a competition. The answer has to be exact, because a remainder that is off by one is simply wrong. This is what I submitted to the SAIR Foundation’s Modular Arithmetic Challenge, why the obvious approach cannot work, and the one idea that made the difference.

The mathematics of the Transformer, worked through from the 2017 paper by Vaswani and colleagues. Scaled dot-product attention, why the scaling factor is there, multi-head attention, positional encoding, and the encoder and decoder stacks, each derived rather than asserted, for a reader who wants to understand the architecture rather than only use it.