| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 6, 2014 |
| Latest Amendment Date: | July 30, 2018 |
| Award Number: | 1402852 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Marian Bocea
mbocea@nsf.gov (703)292-2595 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2014 |
| End Date: | June 30, 2020 (Estimated) |
| Total Intended Award Amount: | $600,000.00 |
| Total Awarded Amount to Date: | $600,000.00 |
| Funds Obligated to Date: |
FY 2015 = $120,000.00 FY 2016 = $120,000.00 FY 2017 = $120,000.00 FY 2018 = $120,000.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
5801 S ELLIS AVE CHICAGO IL US 60637-5418 (773)702-8669 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
5734 S. University Ave Chicago IL US 60637-5418 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
01001718DB NSF RESEARCH & RELATED ACTIVIT 01001516DB NSF RESEARCH & RELATED ACTIVIT 01001819DB NSF RESEARCH & RELATED ACTIVIT 01001415DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The PI is a leading expert in the field of partially hyperbolic dynamics; dynamics is the study of systems, physical or mathematical, that evolve over time according to a deterministic set of rules. Hyperbolic dynamical systems are those which display chaotic, unpredictable features at every point; they are both naturally occurring and well-studied. There are other dynamical systems called KAM systems (named after Kolomogorov, Arnold, and Moser), which have stable regions of regular motion. Partially hyperbolic systems provide a more general class of dynamical systems than either, and include systems that combine hyperbolicity in some directions with KAM behavior in other directions. Partially hyperbolic systems occur widely in dynamical systems arising in physics; for example planetary motion usually contains partially hyperbolic subdynamics, and the effective construction of particle accelerators (used in biological imaging, as well as theoretical physics) requires a detailed understanding of both KAM and partially hyperbolic dynamics. The PI has a well-developed research plan of over 15 years studying partially hyperbolic systems and is poised to raise the theory of these systems to a new level of generality and applicability. The impacts of this research will be seen in future applications to systems in biology, physics and engineering. The PI is currently collaborating with the particle accelerator group at Fermilab to explore some of these potential applications.
The research supported by this grant is guided by the far-reaching goal of developing a general theory of partially hyperbolic systems along the lines of the hyperbolic theory developed in the past 40 years. In particular the PI proposes to study: ergodic properties of conservative partially hyperbolic diffeomorphisms; physical measures for (dissipative) partially hyperbolic diffeomorphisms; rigidity phenomena connected to partially hyperbolic group actions; and ergodicty of singular partially hyperbolic systems. These research goals will be carried out through a variety of modalities, including published papers in peer-reviewed journals, supervising Ph.D. students, and public speaking, both at research conferences and to the general public.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Wilkinson's work centers on asymptotic features -- topological, geometric and probabilistic -- of smooth dynamical systems, in particular diffeomorphisms of closed manifolds. With Keith Burns, she has proved to the most general criterion to date for ergodicity of a conservative dynamical system. The class of diffeomorphisms they consider -- the partially hyperbolic diffeomorphisms -- displays complex dynamical characteristics that are only beginning to be understood. Part of Wilkinson's research program has been to delineate some of these features, including pathological foliations, rigidity of partially hyperbolic actions, and existence of physically significant invariant measures for partially hyperbolic actions. For example, during the award period, with collaborators Artur Avila and Sylvain Crovisier, she has established the C^1 case of a famous conjecture of Pugh and Shub about stable ergodicity of partially hyperbolic systems.
Wilkinson also studies the interplay between dynamics and the algebraic structure of the group of diffeomorphisms, and has proved with her postdoctoral collaborator Jinxin Xue, that certain solvable group actions are highly rigid. In work with Danijela Damjanovic and postdoctoral student Disheng Xu, she discovered a new type of rigidity, called centralizer rigidity, that connects rigidity theory with that of actions of groups.
Much of Wilkinson's work involves the interplay between dynamics and other areas. Her interests thus bring her into close interaction, through seminars and advising, with a broad swath of the graduate population at Chicago.
Last Modified: 06/03/2021
Modified by: Anne M Wilkinson
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