Mathematical Aspects of Materials Science and Prediction with Expert Advice

New York University, New York NY

Investigators

Abstract

This project has two main thrusts. The first is a topic at the interface between mathematics and materials science, involving a class of artificial materials known as "Maxwell lattices." This work will study how the macroscopic mechanical properties of a Maxwell lattice are determined by its microscopic structure. An improved understanding of the relationship between microstructure and macroscopic behavior is expected to facilitate the design of better artificial materials. The project's second thrust lies at the interface between mathematics and data science, and is thus aligned with one of NSF's ten Big Ideas ("Harnessing the Data Revolution"). This work will focus on a problem of sequential decision making, in which data that arrive incrementally must be assembled into a coherent whole despite inconsistencies; the specific focus will be a recently-introduced approach to low-rank matrix completion. In both areas, the project will involve PhD students. The junior scientists involved in this research will gain experience and breadth in both mathematics and a key application area. The project's first thrust will explore the nonlinear mechanics of two-dimensional "Maxwell lattices." These are, by definition, periodic lattice structures in which the average number of edges meeting a node is four; the well-known Kagome lattice is a favorite example. Maxwell lattices are interesting because they are, in most cases, elastically degenerate. A large literature has developed concerning their linearly elastic behavior; however the nonlinear mechanics of these structures is still poorly understood, even for deformations with little or no strain. The project's first thrust will use methods from nonlinear homogenization and the calculus of variations to explore the nonlinear mechanics of Maxwell lattices. The project's second thrust will explore a recently introduced approach to low-rank matrix completion. This approach relies on a two-person, zero-sum, sequential game analogous to the model of online machine learning known as "prediction with expert advice." The principal investigator recently used methods from optimal control to achieve fresh insight concerning prediction with expert advice; it is expected that control-based methods will also be useful for matrix completion. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.