This project addresses several problems of current scientific and technological importance, from fluid mechanics to biology. Specifically, the first project studies the time dependent events that underlie fluid turbulence, events that are too fast to have ever been resolved either in experiments or theory. These events underpin the energy cascade in fluid turbulence, and have immense technological importance. Resolving them is a long-standing problem in fluid mechanics, which is highly related to the question of regularity of the Euler equation. The second topic is a mathematical and computational study of the shape of the nasal cavity, and how it affects the airflow and functionality of the nose. It has long been known that the shape of the nasal cavity changes dramatically across evolution, ranging from dogs with complex labyrinth like internal cavities, to humans who have short internal cavities. This project aims to develop a mathematical framework to understand how the origin of these changes. The third project aims to develop new mathematical methods for aiding drug discovery. Currently there is no efficient way to search the space of small molecules to discover ligands that are relevant for drug targets. Such a search must both find molecules that bind to relevant protein targets and don't bind to unintended targets. Advances in high throughput screening has generated enormous datasets for the interaction between proteins and small molecules. The PI and his collaborators have shown that a method for associating ligands with protein targets based on random matrix theory has higher accuracy than any other published methods, and this project aims to develop this method further to impact the drug discovery pipeline. The broader impact centers around both personnel development of graduate students, undergraduate students and postdocs, and the development of educational materials for teaching science and mathematics through cooking. The teaching initiatives on science and cooking have reached nearly 250,000 people through an online class.

The study of the events underlying fluid turbulence will focus on the collision of two antiparallel vortex filaments. The PI and his collaborators recently published an analysis suggesting that such a collision could lead to a cascade of structures on ever smaller scales, due to the breakdown of a similarity solution of the Biot Savart equations that asymptotically governs the collision. These solutions will be tested both computationally and experimentally in the experimentally realizable problem of the collision of two vortex rings. The research team will numerically track the development of small scale structure, compare them to parallel experiments, and develop a mathematical description of the smallest scales in fluid turbulence. The study of the fluid mechanics of the nasal cavity begins with CT images of noses from different organisms, together with a CFD code for solving the flow field to generate the flows. The research team then will develop an approximate analytical description of the flow through a tortuous cavity that allows understanding of these results. Scaling laws will be developed based on these solutions to arrive at a mathematical description of the design principles. The study of protein ligand binding is centered around deviations from the BBP threshold of the Marcenko Pastur distribution of random matrix theory. By incorporating more information about the chemistry of ligands into this description the research has the potential to improve predictive power for utility in the drug discovery pipeline.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
1715477
Program Officer
Pedro Embid
Project Start
Project End
Budget Start
2017-08-01
Budget End
2021-07-31
Support Year
Fiscal Year
2017
Total Cost
$385,822
Indirect Cost
Name
Harvard University
Department
Type
DUNS #
City
Cambridge
State
MA
Country
United States
Zip Code
02138