Collaborative Research: Topological Methods for the Study of Nonlinear Infinite Dimensional Systems

我们周围的模式会随着时间的推移而改变它们的形状和结构——具体的例子包括：材料中的化学浓度、流体中的速度分布、生态系统中的人口密度等。利用现代信息技术，通过实验或数值模拟收集这些模式的大量数据相对容易。该项目的一个目标是开发和使用计算拓扑技术来使用观察到的模式来识别、量化和分类底层系统的时间相关属性。使用传统的数学技术来分析随时间演化的空间依赖系统的数学模型通常是极其困难的，因此我们对这些系统详细演化的了解大多来自于数值模拟。然而，执行这些数值模拟的过程引入了误差，这些误差可能会增长和传播。原则上，代数拓扑性质在小扰动下保持不变。考虑到这一点，本项目的另一个目标是将新开发的计算拓扑工具与标准数值方法相结合，以验证通过数值模拟获得的解确实是所研究系统的有效结果。

We are surrounded by patterns that change their shape and structure with time -- particular examples include: chemical concentrations in materials, velocity profiles in fluids, population densities in ecosystems, etc. With modern information technologies it is relatively easy to collect enormous data on these patterns either through experimentation or numerical simulation. One goal of this project is to develop and employ computational topological techniques to use the observed patterns to identify, quantify and classify the time-dependent properties of the underlying systems. Mathematical models for spatially dependent systems that evolve with time are typically extremely difficult to analyze using traditional mathematical techniques, and thus much of what we know about the detailed evolution of these systems comes from numerical simulations. However, the process of performing these numerical simulations introduces errors, which can potentially grow and propagate. In principle, algebraic topological properties remain invariant under small perturbations. With this in mind, another goal of this project is to combine the newly developed computational topological tools with standard numerical methods to verify that the solutions obtained through numerical simulation are indeed valid results for the systems being studied.