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. The method combines Conditional Flow Matching, Geometry-Aware Sampling, Reward-Guided Policy updates, and Stochastic Local Search to learn a sampler concentrated near rare, high-quality configurations while keeping generated objects feasible throughout the search. This is especially useful in extremal geometry and combinatorics, where the landscape is highly nonconvex, gradients are unavailable or unreliable, and brute-force enumeration is impossible. The broader message is that, for many problems in experimental mathematics, a generative search with geometric inductive bias can provide a practical alternative to both hand-crafted heuristics and expensive LLM-based systems, suggesting a practical route toward de novo mathematical structure design.