Award Abstract # 1700336
Model Theory and Differential and Difference Equations

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: RESEARCH FOUNDATION OF THE CITY UNIVERSITY OF NEW YORK
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: FY 2017 = $60,726.00
History of Investigator:
  • Joel Nagloo (Principal Investigator)
    jnagloo@uic.edu
Recipient Sponsored Research Office: CUNY Bronx Community College
2155 UNIV AVE
BRONX
NY  US  10453-2804
(718)289-5183
Sponsor Congressional District: 13
Primary Place of Performance: CUNY Bronx Community College
2155 University Ave.
Bronx
NY  US  10453-2804
Primary Place of Performance
Congressional District:
13
Unique Entity Identifier (UEI): C2SFANWGKP39
Parent UEI:
NSF Program(s): FOUNDATIONS
Primary Program Source: 01001718DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):
Program Element Code(s): 126800
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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Blázquez-Sanz, David and Casale, Guy and Freitag, James and Nagloo, Joel "Some functional transcendence results around the Schwarzian differential equation." Annales de la Faculté des sciences de Toulouse : Mathématiques , v.29 , 2020 https://doi.org/10.5802/afst.1661 Citation Details
Casale, Guy and Freitag, James and Nagloo, Joel "Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups" Annals of mathematics , v.192 , 2020 https://doi.org/ Citation Details
Nagloo, Joel "Algebraic independence of generic Painlevé transcendents: PIII and PVI" Bulletin of the London Mathematical Society , v.52 , 2019 https://doi.org/10.1112/blms.12309 Citation Details
Nagloo, Joel "Model Theory and Differential Equations" Notices of the American Mathematical Society , v.68 , 2021 https://doi.org/10.1090/noti2221 Citation Details
Nagloo, Joel and Ovchinnikov, Alexey and Thompson, Peter "Commuting planar polynomial vector fields for conservative Newton systems" Communications in Contemporary Mathematics , 2019 10.1142/S0219199719500251 Citation Details

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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