RTG: Integrating Dynamics and Stochastics (IDyaS)

动力学和随机学都是在最广泛的意义上解释的，都是为应用程序做出了许多贡献的重要数学领域。这两个领域之间有许多天然的联系，这导致人们更好地欣赏在这两个领域中使用的技术和方法。此外，将随机建模与动力系统分析相结合的方法通常最有可能成功地解决问题：三个例子是非线性光学，来自各种来源的噪声在激光和光纤动力学中扮演着重要角色；随机网络，在许多情况下，网络的稳定性可以通过分析由常微分方程或偏微分方程式表征的相关流体极限来确定；以及细胞生理学，其中离子通道门的随机模型通常与实验更吻合。在理论层面上也出现了类似的趋势：概率方法是帮助分析偏微分方程的重要工具，例如，在推导和验证标度定律和非均匀介质中锋面的动力学方面；相反，动力系统方法提供了对有噪声的偏微分方程行为的洞察。这个项目的目标是通过整合动力学、随机学及其应用领域的研究和教育，扩大和提高为研究生和博士后研究员提供的教育和研究培训的范围和质量，并让更多的本科生参与到应用数学的课程和研究经验中。动力系统和随机过程是非常活跃和令人兴奋的研究领域，对经济学和自然科学和社会科学的许多应用做出了重要贡献，同时也具有内在的数学兴趣。动力系统理论研究的是依赖于时间的过程，而随机学则研究的是不确定的随机过程。这两个领域之间的相互作用很重要的例子有：高功率激光设计中的噪声波动，社会网络中形成共识的随机相互作用系统的长期行为，药物输送的微结构和纳米结构的自组装，生物细胞过程中的随机波动，以及金融领域罕见事件的重要性抽样。为研究生和博士后研究员提供更系统和综合的动力学和随机学培训，将使这些群体为在学术界和工业界的职业生涯做好更好的准备。它还将导致接受随机系统培训的美国数学家数量增加，从而使美国能够更好地与欧洲竞争，欧洲在这一领域拥有更大的社区。针对本科生的这些举措将增加接触应用数学并参与暑期研究体验的学生数量。

Dynamics and stochastics, both interpreted in their broadest sense, are important mathematical areas that have made many contributions to applications. There are many natural links between these two areas that lead to a better appreciation of techniques and methods used in either field. In addition, an approach that combines stochastic modelling with a dynamical-systems analysis often has the best chances of tackling a problem successfully: three examples are nonlinear optics, where noise from various sources plays an important role in laser and fiber dynamics; stochastic networks, where stability of the network can, in many cases, be determined by the analysis of a related fluid limit that is characterized by an ordinary or partial differential equation; and cell physiology, where stochastic models of ion channel gates often provide better agreement with experiments. A similar trend occurs on the theoretical level: probabilistic methods are an important tool that aids in the analysis of PDEs, for instance in the derivation and validation of scaling laws and the dynamics of fronts in heterogeneous media; conversely, dynamical-systems methods provide insight into the behavior of PDEs with noise. The goal of this project is to broaden and enhance the scope and quality of the educational and research training provided to graduate students and postdoctoral fellows by integrating research and education in the fields of dynamics, stochastics, and their applications and to involve more undergraduate students in courses and research experiences in applied mathematics.Dynamical systems and stochastic processes are highly active and exciting fields of research that make important contributions to many applications in economics and the natural and social sciences, whilst also being of intrinsic mathematical interest. Dynamical-systems theory is concerned with time-dependent processes, while stochastics deals with nondeterministic, random processes. Examples where the interplay between these two fields is important are noise fluctuations in the design of high-power lasers, the long-time behavior of random interacting systems such as consensus formation in social networks, self-assembly of micro- and nanostructures for drug delivery, random fluctuations in biological cell processes, and importance sampling of rare events in finance. Providing more systematic and integrated training in dynamics and stochastics for graduate students and postdoctoral fellows will prepare these groups better for careers in academia and industry. It will also lead to an increase in the number of US mathematicians trained in stochastic systems, and therefore make the US better able to compete with Europe, which has a larger community in this area. The initiatives for undergraduate students will increase the number of students who are exposed to applied mathematics and are engaged in summer research experiences.