Mohammed Abouzaid
Stanford University, CA·Mathematics & Statistics
Geometry and topology are the branches of mathematics that study shapes and how they bend, connect, and fit together. Two projects approach these fields from different directions.
- Fields
Mathematics & Statistics
Computer Science
- Topics
symplectic geometry
symplectic topology
low-dimensional topology
- Grants
- 2 known grants, through Aug 2029
What they work on
- Train problem-specific neural networks and reinforcement-learning agents to search for constructions in the symplectic ball-packing problem, which asks how efficiently balls fit into a region while preserving signed surface areas.
- Refine and sample near-optimal geometric constructions with diffusion-based generative models, and certify them as rigorous mathematics using interval arithmetic.
- Distill machine-discovered strategies into short, interpretable algorithms that mathematicians can verify and extend, applied to Lagrangian submanifolds and four-dimensional topology.
- Formulate algebraic structures geometrically at the level of flow categories, objects built from Morse theory's gradient paths, to advance Floer homotopy theory.
- Apply the flow-category framework to symplectic topology and to low-dimensional topology through Heegaard-Floer homology.
More about the research
One builds new theoretical machinery from a century-old idea linking shape to functions. The other tests whether artificial intelligence (AI) can help find new constructions and theorems, in an area where today's AI, strong at text, struggles with spatial reasoning.
Why it matters
This is pure mathematics. Its main contribution is new understanding of the structure of geometric spaces, along with general principles for when machine learning can help discover and prove mathematical results. Openly released software and datasets could help other researchers apply AI in mathematics. Lessons are also expected to carry over to real-world problems governed by geometric constraints.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Jun Li
California State University-Northridge, CA·Mathematics & Statistics
Explore whether results on braids and Dehn twists (basic twisting symmetries) along Lagrangian spheres extend to general symplectic 4-manifolds, using surgery on Lefschetz fibrations.
Daniel Freed
Harvard University, MA·Mathematics & Statistics
Investigate expected relationships between Fukaya categories of symplectic resolutions, which are algebraic structures built by counting pseudoholomorphic curves, and structures from representation theory, using microlocal sheaf theory.
Adam Levine
Duke University, NC·Mathematics & Statistics
Investigate knot concordance questions, including homology slice knots, ribbon concordance and piecewise-linear concordance of knots in homology 3-spheres, using Heegaard Floer homology invariants.
The school
Stanford University: 87 labs on PI Pages, $1.6B a year on research, 739 research doctorates a year.
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