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.
2026Algebraic
Hirsch.
The agent builds a linearly presented square-free monomial ideal one generator at a time, searching for a generator graph whose diameter exceeds the ideal’s degree. Valid counterexamples are exceptionally rare, so reward arrives only at the end of a successful construction.
Accepted · 2026GENERATOR GRAPH / GI READY
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.
2026
2025Andrews–
Curtis.
Starting from a balanced presentation, the agent applies Andrews–Curtis moves: relator inversions, multiplications, and conjugations. Each move preserves the underlying group while reshaping the presentation, so progress can require a long sequence of locally valid but strategically meaningful moves.
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 group
The Caltech
Math-AI team.
Support & partnerships
The Lab is grateful to the institutions and partners whose support makes this work possible.
AIMing grant
Foundation
TPU Research Cloud
Academic Grant Program
Academy Research Program
University ProgramLes KohnPhilanthropic support