Osama Khalil
University of Illinois Chicago, IL·Mathematics & Statistics·Early-career
Many deep questions about numbers and shapes can be recast as questions about motion. Picture a system that changes step by step under a fixed rule, then ask what it looks like after a very long time.
- Fields
Mathematics & Statistics
- Topics
homogeneous dynamics
dynamical systems
ergodic theory
- Grants
- 1 known grant, through May 2029
What they work on
- Study how rational points are distributed near self-similar (fractal-like) sets, using homogeneous dynamics, the study of motion on spaces built from symmetry groups, to attack questions in Diophantine approximation.
- Investigate the limits of horocycle-invariant measures, averages that stay unchanged under a shearing motion, on moduli spaces of Abelian differentials, which catalogue flat surfaces of a given type.
- Establish rates of mixing, meaning how fast motion scrambles points, for geodesic flows on negatively curved manifolds, using methods from representation theory, geometry and additive combinatorics.
More about the research
Does it spread out evenly ('equidistribution')? Does it quickly forget where it started ('mixing')? When these systems come from algebra, tools from many branches of mathematics can be brought to bear on them. This project aims to build new methods in this theory of algebraic dynamical systems. It targets central open questions in Diophantine approximation, the study of how well real numbers can be approximated by fractions, and in the geometry of surfaces.
Why it matters
This is pure mathematics. The work could clarify how rational numbers cluster near fractal sets, how measures behave on spaces of flat surfaces, and how quickly motion on curved spaces mixes. These results may settle long-standing questions in Diophantine approximation and surface geometry. The new dynamical tools could eventually inform other problems in number theory and geometry.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Adam Kanigowski
University of Maryland, MD·Mathematics & Statistics
Study how chaotic and statistical properties arise and interact in smooth dynamical systems where some, but not all, Lyapunov exponents (rates at which nearby paths spread apart) are zero.
Alexander Wright
University of Michigan, MI·Mathematics & Statistics·Early-career
Build quasi-convex co-bounded planes in Teichmüller spaces (spaces of hyperbolic shapes on a surface) by gluing together pieces obtained from dynamics, toward convex cocompact surface subgroups of mapping class groups.
Efren Ruiz
University of Hawaii at Manoa, HI·Mathematics & Statistics
Develop new invariants for subshifts of finite type, symbol-sequence dynamical systems, by studying the graph algebras attached to them and how these algebras interact.
The school
University of Illinois Chicago: 42 labs on PI Pages, $532M a year on research, 359 research doctorates a year.
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