Dynamical Systems: Connecting Theory to Applications

动力系统是数学的一个分支，主要研究过程的时间演化，被广泛认为是科学的基本工具。这里的研究旨在将动力系统理论与其潜在的应用联系起来。它将导致该领域的重大进展，大大扩大其范围。它将促进数学与包括生物科学在内的其他科学学科之间的交叉，重点是流行病学和神经科学的数学研究。这些研究项目将为学生和年轻科学家提供培训机会，首席研究员，一位女数学家，将作为榜样，为科学工作队伍的多样性做出贡献。该研究旨在以两种不同的方式将动力系统理论与应用联系起来。一个是通过结合相关性的思想（如随机噪声和有限时间效应），并通过引入空间和时间尺度的经典概念，以提高现有理论的适用性。首席研究员将致力于增加具体的例子池，以促进该领域的实用性。另一个是使用动力系统工具来解决生物科学中出现的问题。本研究有两个具体的主题。一个涉及的及时性的流行病爆发的反应，这个问题是由延迟微分方程定义的无限维动力系统的框架。另一个项目是分析大脑皮层局部回路中兴奋性和抑制性神经元之间的动态相互作用，作为生物学中普遍存在的竞争的范例。该奖项反映了NSF的法定使命，并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，被认为值得支持。

Dynamical systems, a branch of mathematics concerned with the analysis of time evolutions of processes, is widely recognized as a basic tool for science. The research here is aimed at connecting dynamical systems theory to its potential applications. It will lead to significant advances within the field, greatly expanding its scope. It will promote the cross-fertilization between mathematics and other scientific disciplines including the biological sciences, focusing in this research on the mathematics of epidemiology and neuroscience. The research projects will offer training opportunities for students and junior scientists, and the principal investigator, a female mathematician, will serve as role model and contribute to diversity within the scientific work force. The research seeks to connect dynamical systems theory to applications in two different ways. One is to improve the applicability of existing theory by incorporating ideas of relevance (such as random noise and finite-time effects) and by introducing spatial and temporal scales to classical concepts. The principal investigator will work toward increasing the pool of concrete examples to promote the usefulness of the field. The other is to use dynamical systems tools to tackle problems that arise in the biological sciences. There are two specific topics in this research. One concerns the timeliness of the response to epidemic outbreaks; the problem is cast in the framework of infinite dimensional dynamical systems defined by delay differential equations. Another project analyzes, as a paradigm of the competition ubiquitous in biology, the dynamical interaction between excitatory and inhibitory neurons in local circuits of the cerebral cortex.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.