Collaborative Research: AIMing: Machine Learning for Geometry and Topology

This project aims to develop new artificial-intelligence methods for geometry and topology, the branches of mathematics concerned with the shapes of spaces and the ways they bend, connect, and fit together. These fields pose a distinctive challenge for today's AI systems, which reason capably with text and discrete symbols but struggle with the continuous shapes, spatial intuition, and visual reasoning that geometry demands. The work advances national priorities in artificial intelligence by testing and strengthening the reasoning capabilities of modern AI in settings that stress its current limits, with lessons expected to carry over to real-world problems governed by geometric constraints. The project also broadens training and participation in this emerging area: it will support graduate students and postdoctoral researchers, and release openly available software and datasets that lower the threshold for other researchers to apply machine learning in mathematics. This research establishes a systematic framework for applying modern machine learning to open problems in symplectic geometry and low-dimensional topology, organized around three complementary working 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, the topology of four-dimensional spaces, and related structures. Machine-generated constructions will be converted into rigorous mathematics through certification techniques such as interval arithmetic, so that the project's principal outputs are new constructions, conjectures, and theorems. Beyond specific results, the project aims to establish general principles for when and how machine learning can contribute to geometry and topology, and will openly release the software and data underlying its experiments so that other researchers can adapt these techniques to their own problems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.