Svetlana Jitomirskaya
University of California-Berkeley, CA·Mathematics & Statistics
Many systems combine several rhythms whose frequencies never line up exactly. Examples include planets orbiting the sun, population cycles, quasicrystals (ordered but non-repeating solids) and waves in certain materials.
- Fields
Mathematics & Statistics
Physics
- Topics
quasiperiodic operators
quasiperiodic Schrödinger operators
small denominators
- Grants
- 1 known grant, through Jun 2027
What they work on
- Develop non-perturbative Green's function estimates for lattice quasiperiodic operators when the frequency parameter is restricted to a submanifold of a torus.
- Extend Bourgain–Kachkovskiy methods beyond two dimensions to handle multi-particle quasiperiodic operators.
- Construct new classes of space-time quasiperiodic solutions of nonlinear Schrödinger and nonlinear wave equations by removing the dimensional and arithmetic limits of the Craig–Wayne–Bourgain approach.
- Construct full-dimensional KAM tori and prove arithmetic localization results for multi-dimensional quasiperiodic Schrödinger operators.
More about the research
Motion in such systems is called quasiperiodic. Analyzing it requires handling "small denominators": tracking how often and in what pattern a system comes back very close to where it started. For decades the main tool has been Kolmogorov–Arnold–Moser (KAM) theory, but KAM methods are very hard to apply in more than a few dimensions. This collaborative project develops alternative methods that avoid that limit.
Why it matters
This is basic mathematics. It could give rigorous tools for systems with several incommensurate frequencies, which appear in celestial mechanics, population dynamics, quasicrystals and wave equations. Removing current dimensional limits could let mathematical physicists prove results about multi-dimensional quantum and nonlinear wave systems that are now out of reach.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Bjoern Bringmann
Princeton University, NJ·Mathematics & Statistics
Prove that the Gibbs measure of the three-dimensional cubic nonlinear Schrödinger equation is invariant under its dynamics. This would be the first invariance result for a Gibbs measure that is critical with respect to probabilistic scaling.
Wilhelm Schlag
Yale University, CT·Mathematics & Statistics
Determine whether the kink soliton of the phi-4 model is asymptotically stable, extending a long-range scattering theory for potentials that have a threshold resonance.
Gong Chen
Georgia Institute of Technology, GA·Mathematics & Statistics·Early-career
Determine whether multi-solitons, solutions made of several solitary waves at once, are stable and unique in nonlinear dispersive equations, using PDE, harmonic analysis and spectral theory.
The school
University of California-Berkeley: 46 labs on PI Pages, $1.1B a year on research, 825 research doctorates a year.
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