Scott N. Armstrong, New York University.
Scott N. Armstrong (Courant Institute, New York University) and Charles K. Smart (University of Chicago) were awarded the 2017 SIAM Activity Group on Analysis of Partial Differential Equations (SIAM/APDE) Prize for their paper, “Quantitative Stochastic Homogenization of Convex Integral Functionals,” published in Annales Scientifiques de l’École Normale Supérieure
in 2016. They received their awards at the 2017 SIAM Conference on Analysis of Partial Differential Equations, held last December in Baltimore, Md., where Armstrong also gave a talk entitled “Quantitative Stochastic Homogenization by Variational Methods.”
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 2017 prize committee states that Armstrong and Smart’s work “obtained outstanding results and developed fundamental new techniques that have greatly advanced the field and opened the path to further developments.”
Charles K. Smart, University of Chicago.
Scott Armstrong is currently an associate professor at New York University’s Courant Institute of Mathematical Sciences. His research lies at the intersection of probability and analysis, with a recent focus on stochastic homogenization of partial differential equations (PDEs). Charles Smart is an associate professor of mathematics at the University of Chicago. He is particularly interested in the interaction of nonlinear PDEs and probability, in the form of either scaling limits of statistical physics models or homogenization of PDEs with random coefficients.
Both Armstrong and Smart received their Ph.D.s at the University of California, Berkeley. Their collaboration began after graduation with a “just for fun” project about the infinity Laplacian. This project generated a series of papers, and they have been working together ever since.
They responded collectively to our questions.
Why are you excited to be receiving this prize?
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 in the last several years on the topic of quantitative stochastic homogenization, and we hope that 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, who really inspired us.
Can you tell us a bit about the research that won you the prize?
Our research is about the behavior of solutions to certain PDEs with coefficients that are randomly oscillating on very small length scales. These equations model physical properties (like electrical or thermal conductivity) of composite materials. On smaller scales, the solutions behave very erratically since they depend on the equation’s random oscillations; this is very hard to analyze. Researchers want to be able to rigorously prove that on large macroscopic length scales, the solutions behave in a much simpler way because all of 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. There has recently been a lot of focus among researchers on understanding more precisely this homogenization approximation’s ability to describe the real solution. In our 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 gives a rigorous way of implementing a “renormalization group” approach to the problem, and in later work our methods subsequently lead to an essentially optimal quantitative theory for this specific model.
What does your research mean to the public?
We hope that our research, and that of others working on similar topics, will yield new mathematical approaches for understanding other equations with random coefficients as well as similar models of physical systems. There is still much to understand at the level of basic research. The mathematics we have developed also offers a mathematical foundation for the design of numerical algorithms for computing the macroscopic properties of composite materials, which is of practical importance to engineers.
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