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RUI: Variational and Topological Approaches to Complex Dynamical Systems

RUI：复杂动力系统的变分和拓扑方法

基本信息

批准号：
1813752
负责人：
Chad Topaz
金额：
$ 20.25万
依托单位：
Williams College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-15 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1813752&HistoricalAwards=false
关键词：
RUI Variational Topological Approaches Complex

项目摘要

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groups; efforts to join the applied dynamics and topology mathematical communities; and a study of one-of-a-kind, unpublished mathematical manuscripts of John von Neumann.The investigator and his collaborators study two pattern-forming systems: (1) Semi-arid vegetation patterns are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.

本研究主要研究自然界中两个模式形成系统，旨在建立数学框架，捕捉自然规律，并描述系统的行为。第一个系统是半干旱环境中的植被，通常以卫星图像中可见的大规模旋转条纹模式生长。研究人员和他的合作者将创建一类模型，这些模型以基本的生态学原理为基础，并通过地理信息系统（GIS）观测获得信息。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领域的工作。第二个系统涉及集体行为，就像粒子、物体或代理相互作用时出现的行为，从纳米粒子组装到行人群体再到昆虫群。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。研究人员和他的合作者将通过一个称为计算拓扑学的数学领域的透镜来研究集体行为，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合作者网络;从研究到课堂的管道;增强计算基础设施;关注代表性不足的群体的参与;努力加入应用动力学和拓扑数学社区;以及对约翰·冯·诺依曼（John von Neumann）独一无二的未发表数学手稿的研究。研究者和他的合作者研究了两个模式形成系统：（1）半干旱植被格局通常被模拟为反应扩散偏微分方程，其中包括一些子集的地球物理和生态成分的建模者选择。研究人员提出了一种新的模式，植被模式的建模是基于能量平衡的基本生态原理，是由GIS数据的动机和验证。主要的数学贡献是分析了一类非局部约束的优化问题。主要的应用贡献是一个统一的框架，植被格局模拟。(2)代理群体的集体行为可能是复杂的。传统上，调查人员诊断这些行为通过研究时间序列的顺序参数选择先验可能感兴趣。从另一个角度来看，Pl发展了集体行为的拓扑理论，建立了数学机制，从集体行为的模型过渡到更简单的描述，捕捉它们的平均拓扑行为。主要的数学贡献是一个简单的表征复杂的集体动力学，是在不断发展的贝蒂数的系统。主要应用的贡献是一个框架，用于理解集体行为模型，不需要一个先验选择的顺序parameters.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

An unpublished manuscript of John von Neumann on shock waves in boostered detonations: historical context and mathematical analysis

约翰·冯·诺依曼关于助推爆炸冲击波的未发表手稿：历史背景和数学分析

DOI：
10.1007/s00407-020-00258-9
发表时间：
2021
期刊：
Archive for History of Exact Sciences
影响因子：
0.5
作者：
Knoedler, Molly Riley;Kostas, Julianna C.;Hogan, Caroline Mary;Kerkhoff, Harper;Topaz, Chad M.
通讯作者：
Topaz, Chad M.

Connecting the Dots: Discovering the “Shape” of Data

连接点：发现数据的“形状”

DOI：
10.3389/frym.2021.551557
发表时间：
2021
期刊：
Frontiers for Young Minds
影响因子：
0
作者：
Feng, Michelle;Hickok, Abigail;Kureh, Yacoub H.;Porter, Mason A.;Topaz, Chad M.
通讯作者：
Topaz, Chad M.

Analyzing collective motion with machine learning and topology