Mathematical and Computational Studies of Interfaces and Defects

本项目将专注于开发和分析与各种界面和缺陷问题相关的复杂能量图景的数值方法，例如，经历相变的材料的微观结构、几何受阻构型中的量子化涡旋等。他还将研究相关的随机动力学和流体动力学，以及它们的数值近似。他还将学习在基础结构和重要特征的统计分析中有用的稳健和自适应算法。这些研究问题一方面是由与其他科学家合作的实际应用所驱动的，另一方面也推动了从几何和拓扑学到数值和随机分析等数学学科的新研究。将进行的研究是跨学科的，包括计算数学、物理、信息、材料和生物科学等学科。研究中涉及到许多数学和数值挑战，如了解非线性偏微分方程组和随机动力学的家庭集体行为和解的路径，探索模拟结果中的隐藏结构和统计，研究算法开发和数值模拟有助于提高科学计算的能力和预测能力，具有潜在的重大科学、社会和经济影响，是国际上的研究重点之一。同时，界面和缺陷在自然界中也是无处不在的，它们在物理和生物系统的许多方面都起着基础性的作用。对界面和缺陷，特别是在随机环境下的更好的数学和计算理解，将丰富科学知识基础，这反过来可能有助于物理学家、材料科学家和工程师努力发现具有理想性能的新材料，并开发新的科学设备和商业仪器。该项目还将为未来一代劳动力和研究人员提供宝贵的跨学科研究机会。通过强调团队精神(在实验、分析、建模和模拟方面的训练)，年轻的学生可以更好地准备在他们的职业生涯中进行跨学科的研究。

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.