Scott N. Armstrong of the Courant Institute, New York University, and Charles K. Smart of the University of Chicago were awarded the 2017 SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) Prize for their paper, “Quantitative Stochastic Homogenization of Convex Integral Functionals,“ published in 2016 in Annales Scientifiques de l’École Normale Supérieure. Armstrong and Smart received their awards and Armstrong gave a talk, “Quantitative Stochastic Homogenization by Variational Methods,” on December 11, 2017, at the SIAM Conference on Analysis of Partial Differential Equations (PD17) in Baltimore, Maryland.
The SIAG/APDE Prize is awarded biennially to the author or authors of the most outstanding paper, according to the prize committee, published in a peer-reviewed journal in the three calendar years preceding the award year. The prize committee for the 2017 award states that this work by Armstrong and Smart “obtained outstanding results and developed fundamental new techniques which have greatly advanced the field and have opened the path to further developments.”
Both Armstrong and Smart received their PhD, in 2009 and 2010 respectively, at the University of California, Berkeley, where they pursued different fields. Their collaboration started after graduation with a project “just for fun” about the Infinity Laplacian. This generated a series of papers, and they have been collaborating ever since.
Scott Armstrong is currently Associate Professor at the Courant Institute of New York University. His research lies at the intersection of probability and analysis and has been focused recently on stochastic homogenization of partial differential equations.
Charles Smart is Associate Professor of Mathematics at the University of Chicago. His research is focused on nonlinear partial differential equations and probability. He is particularly interested in the interaction of the two, in the form of either scaling limits of statistical physics models or homogenization of PDEs with random coefficients.
They collaborated on their answers to our questions.
Q: Why are you excited to be receiving this prize?
A: We are deeply honored to win this prize and grateful to the committee for acknowledging our work in this way. There has been a lot of interesting mathematics created in the last several years on the topic of our paper, quantitative stochastic homogenization, and we hope the prize shines a spotlight not just on our paper but also on the work of our collaborators Tuomo Kuusi and Jean-Christophe Mourrat as well as the excellent work of others such as Antoine Gloria and Felix Otto and their collaborators, who really inspired us.
Q: Could you tell us a bit about the research that won you the prize?
A: The research is about the behavior of the solutions to certain partial differential equations with coefficients which are random oscillating on very small length scales. These equations model, for instance, physical properties (like electrical or thermal conductivity) of composite materials. On smaller scales, the solutions behave very erratically since they depend on the random oscillations of the equation, and this is very hard to analyze. One wants to be able to prove rigorously that on large macroscopic length scales, the solutions behave in a much simpler way because all this randomness averages out in some sense. This phenomenon is called ``homogenization" and the theory has many similarities to problems in statistical physics and probability theory. Recently, there has been a lot of focus among researchers on more precisely understanding how well this homogenization approximation describes the real solution. In this paper, we introduced some new ideas for obtaining quantitative bounds on the homogenization error, which arose out of the variational formulation of the equations. The variational interpretation of the equations turns out to give a rigorous way to implement a “renormalization group" approach to the problem, and our methods subsequently lead, in later work, to an essentially optimal quantitative theory for this specific model.
Q: What does your research mean to the public?
A: We hope that our research, and that of others working on similar topics, will lead to new mathematical approaches for understanding other equations with random coefficients as well as similar models of physical systems. There is still a lot to be understood at the level of basic research. The mathematics we have developed also gives a mathematical foundation for the design of numerical algorithms for computing the macroscopic properties of composite materials, which is of practical importance to engineers.
Q: What does being a SIAM member mean to you?
A: SIAM plays an indispensable role in promoting applied mathematics, supporting mathematicians and allowing us to communicate our work, keep up-to-date on the latest exciting research, and collaborate with each other.