Closed-Loop Generative Models for Extremal Geometry - Part I

Beyond theorem proving and formal verification, AI can also act as a scientific instrument for exploring the geometry of mathematical solution spaces. A central difficulty in mathematics is to find the right examples, configurations, extremizers of lower/upper bounds, or counterexamples. These are high-dimensional objects that satisfy delicate combinatorial, geometric, or algebraic constraints and reveal the structure of a problem. In this talk, I will present FlowBoost, a closed-loop generative framework for discovering such extremal structures in continuous and constrained optimization problems. This is especially useful in extremal geometry and combinatorics, where the landscape is nonconvex, gradients are unavailable or unreliable, and brute-force enumeration is impossible. The broader message is that, for many problems in mathematics, a generative search with geometric inductive bias can provide a practical alternative to both hand-crafted heuristics, suggesting a practical route toward de novo mathematical structure design. I also discuss how to use this system as a tool for your LLM agent of interest.