Caltech MathAI

CaltechMathAI

We are a team of Math-AI researchers at Caltech, focused on developing AI systems that can tackle hard research-level math problems. Solving challenging mathematical tasks, such as proving or disproving long-standing conjectures, or establishing difficult theorems, often requires discovering intricate, multi-step solutions. Our mission is to use these hard mathematical problems as environments to design new AI algorithms and architectures that can identify rare solutions carrying disproportionately high rewards. In other words, we aspire to be one of the best AI research labs focused on sparse-reward, long-horizon tasks.

A sparse infinite wave graph with isolated orange peaks rising above hills and valleys

The Algebraic Hirsch conjecture.

Rare non-Hirsch ideals constructed across a wide range of degrees.
First successful use of hierarchical RL in commutative algebra.
Hierarchical Reinforcement Learning for Sparse-Reward Search in Commutative Algebra Code
ALGEBRAIC_HIRSCH.ENV
Environment Demo
GENERATOR GRAPH / GI READY
Interactive Algebraic Hirsch generator graphNodes are joined by level-one edges when their labels intersect in d minus one symbols. Orange marks a diameter path; curved off-red edges are irreducible level-two edges.
INITIAL GENERATOR024578Presampled trajectory ready

HOW TO READ IT Nodes connect when |Si ∩ Sj| = d − 1. Orange traces a diameter path; curved off-red arcs mark irreducible level-two edges.

The hierarchical run is presampled: πS builds the line, then πL completes linearity. Uniform search samples distinct generators randomly.

The Andrews–Curtis conjecture.

153 more presentations solved than the prior RL baseline.
550 unsolved examples reduced to 261 equivalence classes.
New datasets
AC-19: 125K · AC-1M: 1.1M
The Two-Hump Problem: Bridging the Difficulty Gap in Mathematical Reinforcement Learning Code & datasetsWhat Makes Math Problems Hard for Reinforcement Learning: A Case Study GitHub
ANDREWS_CURTIS.ENV
INTERACTIVE SANDBOX
BALANCED PRESENTATION / ⟨x,y | r₁,r₂⟩ READY
r₁5 letters
r₂4 letters

TRY IT Apply legal transformations manually, or run the stored agent trajectory back toward ⟨x,y | x,y⟩.

Free reductions happen automatically after every move. This is a small word sandbox, not a general conjecture solver.

The Hadamard maximal determinant problem.

Maximal-determinant records improved, in orders 51 to 119.
+3.11% over the previous record at n = 51.
Exact ±1 certificates released for every record.
New Records for the Hadamard Maximal Determinant Problem in Dimensions 51, 103, 107, 111, 115, and 119 Code & certificates
HADAMARD_RECORDS.MAXDET
New records
|DET| / NN/2 · ORDERS N ≡ 3 (MOD 4)NORMALIZE BY
Largest known determinants relative to the Hadamard bound For every order n congruent to 3 modulo 4 from 7 to 119, the largest determinant on record divided by the Hadamard bound n to the n over two, with our new records at orders 51, 107, 111, 115, and 119 highlighted in orange.

HOW TO READ IT Each point is the largest determinant on record at that order, divided by the chosen upper bound: Hadamard’s nn/2, or Ehlich’s sharper bound for n ≡ 3 (mod 4), which no matrix of these orders is known to meet.

Orange marks our records, with a stem down to the value each one replaced. Hover or focus a point for its exact ratio.

The snake-in-the-box problem.

Longer snakes in every dimension from 9 to 13.
131 inequivalent snakes of length 191 in dimension 9.
Explicit paths released for independent verification.
A Census of New Snake-in-the-Box Records Paths & data ↗
SNAKE.GROW
Playable puzzle
4D HYPERCUBE / TWO CUBESHEAD 0000
Grow a snake in a hypercubeOrange traces your path. Teal edges and outlined vertices show legal moves from the head. Hover or focus a vertex to highlight its edges and neighbors. Crossed vertices are forbidden. Use the coordinate buttons as a keyboard alternative.
Your snakeLegal moveAdjacent edge×Forbidden vertex

Hover or focus a move to see where it leads.

Your transition sequence

The Caltech MathAI team.

Sergei Gukov

Sergei Gukov

Giorgi Butbaia

Giorgi Butbaia

Davide Passaro

Davide Passaro

Michele Tarquini

Michele Tarquini

Lucas Fagan

Lucas Fagan

Justin Tan

Justin Tan

External collaborators.

Paul Orland

Paul Orland

Portrait of Angus Gruen

Angus Gruen

Portrait of Elli Heyes

Elli Heyes

Coco Xiaoyu Huang

Coco Xiaoyu Huang

Maksymilian Manko

Maksymilian Manko

The lab is grateful to the institutions and partners whose support makes this work possible.