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

该项目将广泛应用于物理、化学、材料科学和数据科学中的问题；将使用几个数学学科的想法，包括偏微分方程组、微局部分析和调和分析。最近，这些领域的数学在与量子力学中的分子结构、沉积在表面的原子的动力学、将网络划分为簇状基团、流体动力学中复杂现象的稳定性等方面的各种应用方面取得了长足的进步。这里的研究主要是基于偏微分方程组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计算算法和分析的实现。项目将涉及培训各级研究人员，包括本科生、研究生和博士后学者，以及与统计、物理、计算化学、科学计算和网络科学的研究人员合作。这项研究项目旨在研究几个重要的跨学科课题，从分子结构的基本问题到数据分析中的算法探索。例如，主要研究人员将继续从密度泛函理论出发研究模型中的分叉理论，以及寻找数值和分析工具来研究量子光学中产生的拓扑格子模型的非线性分叉。该项目的很大一部分将与学生和同事合作，开发快速、准确地计算复杂几何形状中的流体流动的数值方法，目的是进行实验预测。将特别关注高度非线性的模型和光谱理论工具的应用。除了物理应用，光谱理论还可以在数据分析中发挥重要作用，特别是在光谱聚类中。沿着这些思路，使用概率、优化和调和分析的思想，研究人员旨在开发网络中的子图检测算法，以及更广泛的马尔可夫链划分。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.