| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 5, 2023 |
| Latest Amendment Date: | May 5, 2023 |
| Award Number: | 2307342 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Stacey Levine
slevine@nsf.gov (703)292-2948 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2023 |
| End Date: | July 31, 2026 (Estimated) |
| Total Intended Award Amount: | $300,000.00 |
| Total Awarded Amount to Date: | $300,000.00 |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
3112 LEE BUILDING COLLEGE PARK MD US 20742-5100 (301)405-6269 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
3112 LEE BLDG 7809 REGENTS DR College Park MD US 20742-5103 |
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Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| NSF Program(s): | APPLIED MATHEMATICS |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Self-organization, that is the emergence of a collective behavior out of the local interactions between members of a group, is ubiquitous in applied sciences. Some bacteria, for example, are attracted toward each other by chemical signals and can form large cohesive clusters that act as a new super-organism. This ability to aggregate is essential to their ability to survive and proliferate. The mathematical description of these bacteria's behavior is similar to that of other self-organizing phenomena, such as the flocking behavior of birds or congested crowd motion, and analogous ideas have been used to model tumor growth. In all cases, cohesive group formation is the result of the competition between long-range attractive and short-range repulsive interactions between the members. The investigator will study the relationship between the one-to-one local interactions and the resulting collective motion for a class of mathematical models that take in consideration these two competing forces. These models are often complex systems of partial differential equations, which describe the motion of individual members or of the members' density distribution function. The goal of this research is to derive, via asymptotic analysis and singular limits, new effective models of geometric type describing the collective motion. And, to use these simpler models to theoretically and numerically study the long time dynamic of a population of bacteria, predict the behavior of a crowd, or compare the effects of different therapies on tumor growth. The project will offer research training opportunity for students.
The investigator will primarily study models for which an interface separating regions of high and low aggregation density can be identified (phase separation). So, while the starting point is a system of partial differential equations that describes the evolution of a density function, the resulting collective motion is modeled by a free boundary approximation describing the evolution of an interface. A rigorous mathematical analysis will be developed using tools from the theory of partial differential equations, the calculus of variation, optimal transportation, and geometric measure theory. A key goal is to provide rigorous justification of the fact that nonlocal attractive behavior has the same smoothing effect on the interface as surface tension (at an appropriate scale). An asymptotic analysis will be performed first on macroscopic models (e.g., diffusion-aggregation equations) and then on mesoscopic models, such as kinetic equations. Understanding how congestion effects can be account for in kinetic models is an important aspect of this research. The investigator will also derive and study free boundary approximations modeling cell motility. The rigorous analysis of these models will establish the instability and symmetry breaking properties, which correspond to well documented behaviors of cells (the so-called self-polarization of cells). Finally, many of the models discussed here have a particular structure: They are gradient flows with respect to the Wasserstein distance - which is defined via the theory of optimal transportation. The investigator will pursue the development of a regularity theory for optimal transportation in a discrete setting. This is an important step toward developing effective numerical methods in the field of optimal transportation, with application to the models discussed above.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
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