Mathematical and Computational Studies of Interfaces and Defects

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

This project will focus on the development and analysis of numerical methods to probe the complex energy landscapes related to various interface and defect problems, such as microstructures in materials undergoing phase transitions, quantized vortices in geometrically frustrated configurations, and so on. The associated stochastic dynamics and hydrodynamics, as well as their numerical approximations will also be investigated. He will also study robust and adaptive algorithms that are useful in the statistical analysis of the underlying structures and important features. These research issues, on one hand, are driven by practical applications through collaborations with other scientists, and on the other hand, also motivate new studies of mathematical subjects ranging from geometry and topology to numerical and stochastic analysis. The research to be carried out is of interdisciplinary nature, encompassing subjects like computational mathematics, physics, information, materials and biological sciences. There are many mathematical and numerical challenges involved in the research such as the understanding of the collective behaviors of families and paths of solutions of nonlinear PDEs and stochastic dynamics, and the exploration of the hidden structures and statistics in the simulated results. The research on the algorithmic development and numerical simulations can help enhancing the capability as well as the predictive power of scientific computations, which has potentially significant scientific, social and economic impact and is one of the top research priorities internationally. Meanwhile, interfaces and defects are also ubiquitous in nature which play fundamental roles in many aspects of physical and biological systems. A better mathematical and computational understanding of interfaces and defects, especially in a stochastic setting, will enrich the scientific knowledge base, which in turn may aid the efforts by physicists, materials scientists and engineers in discovering new materials with desirable properties and in developing new scientific devices and commercial instruments. This project will also provide valuable interdisciplinary research opportunities for the future generation of workforce and researchers. With an emphasis on the TEAMS (Training in Experiments, Analysis, Modeling and Simulations) spirit, young students can be better prepared to conduct interdisciplinary research in their career.