Fano varieties — projective varieties with ample anticanonical bundle — are the building blocks of algebraic geometry's classification program. In dimension four, the classification of Fano hypersurfaces with terminal singularities has remained incomplete despite decades of work. The problem is combinatorial: each candidate is specified by a high-dimensional integer vector (the weights and degree of the weighted projective space), and checking the Fano and terminal-singularity conditions requires evaluating algebraic-geometric criteria at each point. The search space is enormous, rewards are sparse, and known combinatorial methods can't reach large regions.
Truter trains a deep reinforcement learning agent to navigate this integer lattice. The agent learns a heuristic that guides exploration toward regions with denser rewards — regions where the algebraic-geometric conditions are more likely to be satisfied. The neural network doesn't prove theorems; it learns where to look.
The result: thousands of previously unknown Fano 4-fold hypersurfaces, including hundreds that are inaccessible to known search methods. The combinatorial obstructions that blocked classical enumeration don't block the RL agent because the agent doesn't follow the same search paths. It learns a different route through the lattice, one shaped by the reward signal rather than by the combinatorial structure that humans exploit.
The agent discovers mathematical objects, not patterns in data. Each discovered Fano variety is checkable — the algebraic-geometric conditions are decidable — so the RL agent proposes candidates and exact verification confirms or rejects them. The learning is heuristic. The mathematics is exact.