This $388,536 project grant awarded by the National Science Foundation (NSF) Division of Mathematical Sciences, under the Mathematical and Physical Sciences (CFDA 47.049) program, supports fundamental research at the University of Massachusetts (UMass) to study the propagation of randomness in nonlinear wave phenomena. The key objectives of the 3-year project are to: Analyze the long-term dynamics and stability of dispersive flows from a probabilistic perspective in energy subcritical regimes....
This $100,972 federal Project Grant award from the National Science Foundation's (NSF) Mathematical and Physical Sciences Program (CFDA 47.049) supports fundamental research on nonlinear partial differential equations and their applications in fields such as crystal growth, combustion, and game theory. The primary investigator (PI) will study the regularity, large-time behavior, and qualitative properties of solutions to these equations, which have connections to areas like the calculus of...
This $300,000 Project Grant award from the National Science Foundation's (NSF) Mathematical and Physical Sciences program (CFDA 47.049) supports fundamental research aimed at advancing the mathematical analysis of nonlinear partial differential equations. The research has three main focus areas: (1) studying fluid dynamics problems with free boundaries, such as water waves, tsunamis, and hurricanes; (2) investigating the dynamics of gases and plasmas under physical kinetic boundary conditions,...
This $142,013 federal Project Grant award from the National Science Foundation's (NSF) Mathematical and Physical Sciences (MPS) program (CFDA 47.049) supports research conducted by New York University (NYU) on the development of new mathematical methods for analyzing complex physical systems with behaviors across multiple length scales. The primary goals of this 3-year project are to: 1) improve quantitative homogenization theory to better account for key parameters and allow for more flexible...
This three-year National Science Foundation Project Grant of $197,999 supports research and education activities at Worcester Polytechnic Institute under the Mathematical and Physical Sciences program (CFDA 47.049). The award will fund the development of novel mathematical methods to analyze partial differential equations describing singularities in continuum mechanics and materials science systems. The principal investigator will extend differential inclusion theory on rigidity and flexibility,...
This $176,355 Project Grant award from the National Science Foundation's (NSF) Mathematical and Physical Sciences (CFDA 47.049) program supports fundamental research on dispersive and wave equations at the University of Chicago. The project aims to advance the mathematical theory of wave turbulence and the analysis of Gibbs measures for Hamiltonian systems, with applications to fields such as plasma physics, nonlinear optics, and quantum field theory. Key research activities include extending...
This Project Grant award of $300,000.00 from the National Science Foundation's (NSF) Mathematical and Physical Sciences Program (CFDA 47.049) supports collaborative research by Princeton University to develop a theoretical framework for understanding the non-linear response of quantum materials. The research aims to establish predictions for out-of-equilibrium behavior in systems like superconductors and thin wires, and to characterize phenomena like disorder-driven localization and chaos in...
The National Science Foundation (NSF) awarded a $299,213 Project Grant under the Mathematical and Physical Sciences (CFDA 47.049) federal grant program to The Trustees of Princeton University to study the dynamics and stability of solutions to important partial differential equations that describe classical systems in applied mathematics and physics, such as incompressible fluids, geophysical flows, plasma, and water waves. The 3-year award, effective July 1, 2024, will establish...
This $210,000 Project Grant award, provided by the National Science Foundation (NSF) under the Mathematical and Physical Sciences program (CFDA 47.049), supports research on nonlinear partial differential equations, geometry, and convex analysis. The research aims to develop new analytical techniques with applications in geometry, physics, and algebraic geometry. Key focus areas include studying special Lagrangian submanifolds, Kähler-Einstein metrics, and Monge-Ampère type equations. The...
This $265,813 National Science Foundation project grant supports collaborative research on singularities in incompressible flows through computer-assisted proofs and physics-informed neural networks. Funded under the Mathematical and Physical Sciences program (CFDA 47.049), the award brings together five principal investigators from leading U.S. research universities to investigate three specific projects: non-uniqueness of Leray-Hopf solutions to the Navier-Stokes equations in three dimensions;...
This $254,498 Project Grant award from the National Science Foundation's (NSF) Mathematical and Physical Sciences program (CFDA 47.049) supports three research projects at Princeton University that advance the mathematical understanding of Gibbs measures and their dynamics. The principal investigator and collaborators aim to:
Prove the invariance of the Gibbs measure for the cubic nonlinear Schrödinger equation in three dimensions, which represents a critical result.
Establish the well-posedness of the stochastic wave maps equation, combining techniques from partial differential equations, probability theory, and differential geometry.
Demonstrate the global well-posedness of the stochastic Yang-Mills-Higgs heat flow in two dimensions, enabling the rigorous construction of two-dimensional Yang-Mills-Higgs measures for Euclidean quantum field theory.
These interconnected projects forge new connections between partial differential equations, probability theory, and differential geometry, advancing mathematical understanding of complex physical phenomena. The award supports fundamental research as well as the training of undergraduate and graduate students, strengthening the STEM workforce in the United States.