This project concerns free boundary surfaces, interfaces that minimize some kind of energy or cost. Two examples are the surface of least area dividing a solid into two halves of equal volume and the first nodal set, the surface where the lowest energy eigenfunction changes sign. The project considers at the same time abrupt transitions from one phase or material to another and more gradual transitions. A main goal is to study the interaction of all levels of the transition and to establish that the transition happens in the simplest possible way. The proposal suggests large-scale geometric mechanisms that cause optimizing level surfaces to resemble parallel planes at smaller scales. A second aspect of the project is to study the fluctuations of a lattice growth model known as internal diffusion-limited aggregation (internal DLA). In the limit, as the lattice size tends to zero, this process approaches a continuum free boundary evolution known as Hele-Shaw flow. In a first, key step, the principal investigator and his coauthors, Lionel Levine and Scott Sheffield, have shown that the fluctuations are closely related to the Gaussian Free Field, a random distribution in two or more dimensions that is analogous to Brownian motion in one dimension.
The interfaces treated in this proposal arise as profiles of wakes and jets, as the boundary surface between oil and water or between ice and water, and in other phase transitions in physics. In each of those cases, the optimal shape is dictated by a physical principle. There are, moreover, examples of interfaces governed by economic principles, such as the surface where an insulating material meets the outside air, optimized to minimize the total cost of the insulating material and the energy needed to maintain a desired temperature. The probabilistic lattice model mentioned above was proposed in the 1980s by chemists Paul Meakin and John M. Deutch to describe the movement of molecules in industrial chemical processes such as electropolishing, etching, and corrosion. The fluctuations studied in this project are the very issue of concern to chemists, who want to estimate the roughness of metal surfaces at the molecular scale. In addition to conducting his own research, the principal investigator organizes and supervises graduate student mentors of undergraduates and high school students in two MIT summer programs, one of which is the internationally recognized Research Science Institute (RSI).
