This project concerns two mathematical research directions. The first direction involves the mathematical study of waves such as water waves or sound waves. Both physical experiments and numerical simulations exhibit the phenomenon of turbulence: fluids that are initially very smoothly-flowing and slowly-varying can develop much more complicated eddies and fine-scale behavior. It remains unknown whether the equations that model fluid dynamics can give rise to "blow-up," in which some portion of the simulated fluid achieves infinite velocity. This project studies the extent to which blow-up can be "engineered" by tweaking the equations that model fluid mechanics, such as changing the number of dimensions in space. The second direction involves questions related to the notorious twin prime conjecture in number theory. This conjecture asserts that there are infinitely many pairs of prime numbers that are separated by a distance of two, such as 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. It is still not known whether the assertion is true, but recently much progress has been made on understanding more tractable questions, such as how often it occurs that a pair of numbers, separated by a distance of two, both have an odd number of prime factors. This project aims to develop these promising new techniques further, with potential application to proving or disproving the twin primes conjecture and to other difficult, important questions in number theory.
In more detail, the fluid dynamical part of the project focuses primarily on variants of the Euler equations for incompressible fluids, especially higher-dimensional Euler equations on Riemannian manifolds. The ability to select the metric of such a manifold gives a promising way to "program" the equations to exhibit certain desirable behavior. Prior work has established that the dynamics of certain quadratic ordinary differential equations can be programmed into such systems. The project will continue work towards exhibiting finite-time blow-up (or other interesting behavior, such as Turing universality) for these models. For the number-theoretic aspects of the project, the investigator and collaborators aim to establish further cases of the Chowla conjecture (or its logarithmically-averaged variants) on correlations of the Liouville function, by combining the recently-developed entropy decrement method with techniques from analytic number theory, combinatorics, and ergodic theory.
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.
