Algorithms and Analysis for Models in Materials Science, Fluids, and Probability

该项目将涵盖物理、化学、材料科学和数据科学问题的应用；将使用几个数学学科的思想，包括偏微分方程、微局部分析和谐波分析。这些领域的数学最近在与量子力学中的分子结构、表面原子沉积动力学、将网络划分为簇群、流体动力学中复杂现象的稳定性等相关的各种应用上取得了巨大的进步。这里的研究主要基于偏微分方程和概率论的新发展理论，尽管工作的几个组成部分将致力于实现计算算法和分析。项目将涉及培养各级研究人员，包括本科生、研究生和博士后学者，以及与统计学、物理学、计算化学、科学计算和网络科学等领域的研究人员合作。该研究项目旨在研究几个重要的跨学科主题，从分子结构的基本问题到数据分析算法的探索。例如，首席研究员将继续研究密度泛函理论模型中的分岔理论，以及寻找用于研究量子光学中出现的拓扑晶格模型的非线性分岔的数值和分析工具。该项目的大部分工作将与学生和同事合作，开发快速、准确计算复杂几何形状中流体流动的数值方法，目的是进行实验预测。将特别注意高度非线性模型和光谱理论工具的应用。除了物理应用之外，光谱理论在数据分析中也可以发挥重要作用，特别是在光谱聚类中。沿着这些路线，使用概率，优化和谐波分析的思想，研究者的目标是开发网络中的子图检测算法，以及更普遍的马尔可夫链划分。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project will span applications to problems in physics, chemistry, materials science, and data science; ideas from several mathematical disciplines, including partial differential equations, microlocal analysis, and harmonic analysis, will be used. Mathematics from these areas has made great strides recently on a variety of applications related to molecular structure in quantum mechanics, dynamics of atoms depositing on a surface, partitioning networks into clustered groups, stability of complex phenomena in fluid dynamics, and more. The research here is based largely on newly-developed theories from partial differential equations and probability, though several components of the work will be devoted to implementation of computational algorithms and analysis. Projects will involve training researchers at all levels, including undergraduates, graduate students, and postdoctoral scholars, as well as collaboration with researchers from statistics, physics, computational chemistry, scientific computing, and network science. This research project is aimed at the study of several important interdisciplinary topics ranging from fundamental questions about molecular structure to explorations of algorithms in data analysis. For instance, the principal investigator will continue work on bifurcation theory in models from density functional theory, as well as on finding numerical and analytic tools for studying nonlinear bifurcations of topological lattice models arising in quantum optics. A substantial part of the project will be done in collaboration with students and colleagues developing numerical methods for fast, accurate computation of fluid flows in complicated geometries, with the intention of making experimental predictions. Special attention will be paid to highly nonlinear models and applications of tools from spectral theory. Beyond physical applications, spectral theory can also play important roles in data analysis, specifically in spectral clustering. Along these lines, using ideas from probability, optimization and harmonic analysis, the investigator aims to develop algorithms for sub-graph detection in networks, and more generally Markov chain partitioning.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.