Project Grant 2617223

The National Science Foundation Division of Mathematical Sciences awarded The Leland Stanford Junior University $333,333 on September 1, 2026, for collaborative research developing machine learning methods for geometry and topology under the Mathematical and Physical Sciences program (CFDA 47.049).

The project establishes a systematic framework for applying modern machine learning to open problems in symplectic geometry and low-dimensional topology. Work centers on three complementary modes: training problem-specific neural networks and reinforcement-learning agents to search for geometric constructions; refining and sampling near-optimal constructions using diffusion-based generative models; and distilling machine-discovered strategies into short, interpretable algorithms that mathematicians can verify, generalize, and extend. The symplectic ball-packing problem—determining how efficiently balls can be embedded into a given region while preserving signed areas of surfaces—serves as a central testbed, alongside questions about Lagrangian submanifolds, four-dimensional topology, and related structures. Machine-generated constructions will be converted into rigorous mathematics through certification processes. The research addresses a distinctive challenge for current AI systems: while they reason capably with text and discrete symbols, they struggle with the continuous shapes, spatial intuition, and visual reasoning that geometry demands. The project supports graduate students and postdoctoral researchers and releases openly available software and datasets to lower barriers for other researchers applying machine learning in mathematics.

Performance occurs at Palo Alto, California, with a period of performance from September 1, 2026, through August 31, 2029.