Jeremy Kahn
Brown University, RI·Mathematics & Statistics
Symmetry comes in two kinds. Discrete symmetry is like the repeating pattern of a crystal.
- Fields
Mathematics & Statistics
Computer Science
- Topics
Lie groups
lattices
surface subgroups
- Grants
- 2 known grants, through Jul 2029
What they work on
- Determine whether three or four definitions of a surface subgroup being "closely aligned" with an sl_2(R) subgroup, inside lattices of semi-simple Lie groups, are mathematically equivalent.
- Prove that complete assemblies of good pants groups, simple surface pieces glued with good matching, yield closely aligned surface subgroups, using good pants homology from the proof of the Ehrenpreis conjecture.
- Adapt the team's rule-based Euclidean geometry theorem prover to the axioms of hyperbolic geometry, a curved geometry, as the logical core of a new reasoning system.
- Generate a large dataset of hyperbolic geometry statements and proofs, and use it to train the AI part of a neuro-symbolic theorem-proving engine.
More about the research
Smooth symmetry is like the continuous rotations of a sphere. Mathematicians describe these with groups: discrete groups for crystal-like patterns and Lie groups for smooth symmetries. This research asks how the symmetries of simple two-dimensional surfaces sit inside the symmetries of larger, more complicated spaces. A second line of work builds artificial intelligence (AI) systems that can prove theorems in hyperbolic geometry, a curved geometry in which parallel lines spread apart. Both projects involve geometry, and both use computers to make mathematical structure easier to see and reason about.
Why it matters
The surface-subgroup work is basic mathematics. It clarifies how simple surface symmetries fit inside complex symmetry groups and could inform geometry and the study of symmetry more broadly. The automated reasoning work aims to give mathematicians, educators and students open AI tools that prove theorems in hyperbolic geometry, a geometry with uses in physics and computer science. It is a step toward AI assistance across research-level mathematics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Matthew Stover
Temple University, PA·Mathematics & Statistics
Use ideas from geometry, topology, algebraic geometry and dynamics to answer open questions, including basic existence problems, about real and complex hyperbolic lattices, which are discrete symmetry groups of curved spaces.
Richard Canary
University of Michigan, MI·Mathematics & Statistics
Study the metric completion of the Hitchin component, a higher Teichmüller space of surface group representations, with respect to a pressure metric that generalizes the Weil-Petersson metric.
Xiaoyu Huang
Temple University, PA·Mathematics & Statistics
Generate and curate large public datasets of number-theoretic and combinatorial objects, including a database of noncongruence modular forms for studying their arithmetic properties.
The school
Brown University: 23 labs on PI Pages, $375M a year on research, 251 research doctorates a year.
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