RUI: Variational and Topological Approaches to Complex Dynamical Systems

本研究研究了自然界中的两个模式形成系统，旨在创建捕捉自然规律并表征系统行为的数学框架。第一个系统是半干旱环境中的植被，它们通常以卫星图像中可见的大规模旋转条纹模式生长。研究者和他的合作者将创建一类模型，这些模型以基本生态学原理为基础，并以地理信息系统（GIS）观测为依据。他们将分析这些模型，并将其与数据进行比较。这项工作将提供深入了解我们的自然世界在植被生态学领域的工作。第二个系统涉及集体行为，比如当粒子、物体或媒介相互作用时产生的行为，从纳米粒子组装到行人人群再到昆虫群。这些群体的行为很难描述，因为它们既不是高度组织化的，也不是完全随机的。研究者和他的合作者将通过一个被称为计算拓扑的数学领域来研究集体行为，该领域检查从这些群体的模拟或实验中获得的数据的形状。这项工作将有助于制定帮助人类理解复杂数据的策略。该提案的其他特点包括：广泛的本科生参与；在四个机构之间建立协作网络；从研究到课堂的管道；加强计算基础设施；注重代表性不足群体的参与；努力加入应用动力学和拓扑数学社区；以及对约翰·冯·诺伊曼独一无二的、未发表的数学手稿的研究。研究者和他的合作者研究了两种模式形成系统：(1)半干旱植被模式通常被建模为反应-扩散PDE，其中包含建模者选择的一些地球物理和生态成分子集。研究者提出了一种基于能量平衡基本生态原理的植被模式建模新范式，该范式受到GIS数据的启发和验证。主要的数学贡献是分析了一类具有非局部约束的优化问题。主要的应用贡献是为植被格局建模提供了一个统一的框架。(2) agent群体的集体行为可能是复杂的。传统上，研究人员通过研究先验选择可能感兴趣的顺序参数的时间序列来诊断这些行为。采用不同的观点，Pl开发了集体行为的拓扑理论，建立了从集体行为模型到捕获其平均拓扑行为的更简单描述的数学机制。主要的数学贡献是对复杂的集体动力学进行了更简单的描述，这种描述是根据系统中不断变化的贝蒂数来构建的。主要的应用贡献是一个框架，用于理解不需要先验地选择顺序参数的集体行为模型。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。

This research studies two pattern-forming systems in the natural world, aiming to create mathematical frameworks that capture natural laws and to characterize the behavior of the systems. The first system is vegetation in semi-arid environments, which often grow in large-scale, swirling striped patterns visible in satellite imagery. The investigator and his collaborators will create a class of models that are grounded in fundamental ecological principles and are informed by geographic information systems (GIS) observations. They will analyze these models and compare them to data. This work will provide insight into the working of our natural world in the area of vegetation ecology. The second system involves collective behaviors, like those that arise when particles, objects, or agents interact, ranging from nanoparticle assembly to pedestrian crowds to insect swarms. The behavior of these groups can be vexing to characterize as they are neither highly organized nor totally random. The investigator and his collaborators will study collective behavior through the lens of a mathematical area called computational topology, which examines the shape of data obtained from simulations of or experiments on these groups. This work will aid the development of strategies that help humanity understand complex data. Other features of this proposal include: extensive undergraduate student involvement; creation of a network of collaborators across four institutions; a pipeline from research to the classroom; enhancement of computational infrastructure; a focus on the participation of underrepresented groups; efforts to join the applied dynamics and topology mathematical communities; and a study of one-of-a-kind, unpublished mathematical manuscripts of John von Neumann.The investigator and his collaborators study two pattern-forming systems: (1) Semi-arid vegetation patterns are commonly modeled as reaction-diffusion PDE that incorporate some subset of geophysical and ecological ingredients selected by the modeler. The investigator proposes a new paradigm for modeling vegetation patterns that is based on the fundamental ecological principle of energy balance and is both motivated by and validated with GIS data. The primary mathematical contribution is the analysis of a class of optimization problems with nonlocal constraints. The primary applied contribution is a unified framework for vegetation pattern modeling. (2) The collective behaviors of groups of agents can be complex. Traditionally, investigators diagnose these behaviors by studying time series of order parameters chosen a priori to be of possible interest. Taking a different view, the Pl develops a topological theory of collective behavior, building the mathematical machinery to pass from models of collective behavior to simpler descriptions capturing their on-average topological behavior. The primary mathematical contribution is a simpler characterization of the complex collective dynamics that is framed in terms of the evolving Betti numbers of the system. The primary applied contribution is a framework for understanding collective behavior models that does not require an a priori choice of order parameters.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.