
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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Initial Amendment Date: | June 7, 2017 |
Latest Amendment Date: | June 7, 2017 |
Award Number: | 1700336 |
Award Instrument: | Standard Grant |
Program Manager: |
Tomek Bartoszynski
tbartosz@nsf.gov (703)292-4885 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
Start Date: | July 1, 2017 |
End Date: | February 28, 2021 (Estimated) |
Total Intended Award Amount: | $60,726.00 |
Total Awarded Amount to Date: | $60,726.00 |
Funds Obligated to Date: |
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History of Investigator: |
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Recipient Sponsored Research Office: |
2155 UNIV AVE BRONX NY US 10453-2804 (718)289-5183 |
Sponsor Congressional District: |
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Primary Place of Performance: |
2155 University Ave. Bronx NY US 10453-2804 |
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): | FOUNDATIONS |
Primary Program Source: |
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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
In this project, the investigator will use model theory, a branch of mathematical logic, to classify and study some classical differential and difference equations. Model theory is a branch of mathematical logic that studies mathematical structures and their definable sets (i.e., the basic sets that "live" in those structures). On the other hand, differential and difference equations are equations that describe how things change with time and are the fundamental tools that modern science and engineering use to model physical reality. The basic idea of the project will be to view the set of solutions of a differential or difference equation as a definable set in an appropriate structure, and use the tools of model theory to understand its properties. The equations that the project focuses on -- the Painlevé differential and difference equations and the Schwarzian differential equations -- appear in many important physical applications such as statistical mechanics and general relativity, as well as in important problems in number theory.
The project builds on recent successes by model theorists to use techniques from geometric stability theory to classify special cases of these differential equations according to the trichotomy theorem in differentially closed fields of characteristic zero. The investigator will extend his work on the generic Painlevé (differential) equations to non-generic ones as well as to the Schwarzian equations. The objective is to both fully describe the structure of the sets of solutions and to use such descriptions to better our understanding of geometrically trivial sets in the theory. This work is expected to have applications to problems in number theory and functional transcendence theory. The project will also transport some of the ideas and techniques used to study classical differential equations to the difference setting. The main challenge will be to develop a notion of irreducibility (in the sense of classical functions) for difference equation and show that the difference Painlevé equations are irreducible. In addition to model theory, the work will also employ techniques from algebra and geometry and from the analytic study of differential/difference equations.
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.
Intellectual merit: During the course of this project, the PI made significant progress on problems that are at the interface of model theory, differential equations and functional transcendence. Notably, in joint work with G. Casale and J. Freitag, the PI proved the Ax-Lindemann-Weierstrass (ALW) Theorem with derivatives for the Fuchsian automorphic functions. Furthermore, this ALW theorem was used to prove certain cases of the André-Pink conjecture in number theory. This work has initiated a new direction of research complementary to those based around the work of J. Pila and A. Wilkie in the context of o-minimality. Moreover, the PI also gave a proof of the algebraic independence conjecture for the generic Painlevé transcendents, completing the work started with A. Pillay in 2011. These results, including several other work, were published in some of the leading journals in mathematics.
Broader impact: The PI gave a number of research talks, including twelve invited conference talks and seventeen research seminar talks. The project has also led to new and ongoing interdisciplinary collaborations. Finally, the PI has also actively organized or served on the committees of many scientific meetings and has provided financial support to graduate students and postdocs for short visits, including to speak/attend some of those meetings.
Last Modified: 06/22/2021
Modified by: Joel Nagloo
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