This project yielded rigorous mathematical theorems about equations with physical origins, including Euler's equations, which describe the motion of fluids such as air and water, and Einstein's equations of General Relativity, which describe the motion of gravitational waves and which are tied to three out of the last five Nobel Prizes in Physics. The project focused on fascinating phenomena in our physical world that are described by the appearance of infinities in the underlying mathematical equations that describe them. Loosely speaking, the infinities are referred to as "singularities." One outstanding example is the Big Bang, which is a singularity in General Relativity such that the curvature of spacetime is infinite. A second important example is shock waves, which are disturbances in a medium caused by sharp (discontinuous) changes in some physical quantity such as density or electric field strength; the "sharpness" corresponds to singularities in the rate of change of the underlying physical quantities and is associated with physical phenomena such as sonic booms and earthquakes. Despite the clear physical importance of these singularities, before the project, there were very few rigorous mathematical theorems that describe or predict their formation and structure. The project led to 16 published papers on these topics, and additional papers are in progress.

The work on fluid motion has revealed new insights about sound waves, shock waves without swirling motion, and more recently, shock waves in the presence of swirling motion, a notoriously complex phenomenon. The works on gravitational waves have led to a deeper understanding of the Big Bang and the long-term fate of the universe. A common theme unifying these evolutionary phenomena is that they involve wave-like motion. The project has yielded new tools for the study of waves, shaped by ideas from geometry. Some of the recent papers have shown that the equations of fluid motion have some unexpected, remarkable commonalities with Einstein's equations. This allowed for the blending of insights and techniques from two seemingly separate fields, which in turn served as a driving force behind the research.

Specifically, for Einstein's equations, the project has provided the first general theorems that describe the dynamic stability of the Big Bang. This means that a rigorous proof was given showing that the Big Bang is a stable prediction (as opposed to a fluke) of General Relativity, under certain assumptions on the present state of the universe and the kind of matter present. In addition, the mathematical tools developed in studying the Big Bang are robust and have applications to other equations of mathematical interest. For Euler's equations, the project provided the first general theorems that describe how shock waves form in contexts where the fluid is swirling. Related results were proved for other equations of mathematical interest.

The project supported a pair of two-week summer schools at MIT. The schools addressed topics lying at the heart of the research results produced by the project. The primary purpose of the schools was to soften the entry barrier into the field by immersing the students in an environment where they experience how different branches of mathematics can be combined to solve important problems. The schools trained more than 70 undergraduates, graduate students, and postdoctoral researchers. They were inclusive of groups traditionally underrepresented in STEM. In particular, the schools led to the recruitment of an underrepresented group member into the field who completed their PhD and went on to win an NSF Postdoctoral Fellowship. The project also supported undergraduate research, provided training for PhD students, and led to the development of new curriculum materials on advanced research topics.

Last Modified: 07/24/2022
Modified by: Jared R Speck