Ergodic and Statistical Phenomena in Dynamics

该项目由两部分组成。在第一部分中，主要研究人员将致力于与流体动力学方程和其他非线性偏微分方程解的奇点出现有关的各种问题。在数学中，这与一个著名的粘土问题有关。PI已经在该地区取得了一些成果。这是借助于所谓的重整化群方法得到的，重整化群方法是动力系统理论中的主要方法之一，它允许人们研究解的各种奇性。该项目将考虑新类型的边界问题和其他类型的非线性偏微分方程组。该项目的这一部分可能还需要一些数值研究。该项目的第二部分将集中于与数论相关的遍历理论中的几个问题。这包括小尺度上线性形式的值的统计，以及与著名的Littlewood猜想和大的Frobenius数的分布有关的问题。必须强调的是，在这种情况下出现的系综和概率分布与传统概率论中的集合和概率分布有很大的不同，需要新的想法和新的方法。该项目的第一个主要部分涉及偏微分方程组，它是提供物理现象的数学模型的主要机制之一。这个项目特别感兴趣的是流体力学中出现的方程，这些方程适用于研究飓风和龙卷风等创伤性事件。该项目的第二个关键部分侧重于遍历理论。遍历理论是概率和动力系统学科结合在一起的一个数学领域。它也应用于真实世界的现象，天气、气候变化和地震的预测，仅举几例。遍历理论通常与数学的许多重要部分一起发展。预计这个项目将在遍历理论方面取得的进展将对我们理解自然界中的许多重要事件产生影响，并可能在物理和地质学等其他科学学科中找到应用。

The project consists of two parts. In the first part, the principal investigator will work on various questions related to the appearance of singularities in solutions of the equations of fluid dynamics and other nonlinear partial differential equations. In mathematics this is connected with one of the famous Clay problems. The PI has already established some results in the area. These were obtained with the help of the so-called renormalization group method, which is one of the main methods in the theory of dynamical systems and allows one to study various kinds of singularities of solutions. The project will consider new types of boundary problems and other types of nonlinear partial differential equations. This part of the project might also require some numerical studies. The second part of the project will focus on several problems from ergodic theory related to number theory. This includes the statistics of values of linear forms on small scales and problems related to the famous Littlewood conjecture and the distribution of large Frobenius numbers. One must stress that the ensembles and probability distributions that arise in this context are quite different from the ones in traditional probability theory and require new ideas and new methods.The first major component of the project deals with partial differential equations, which represent one of the principal mechanisms for providing mathematical models of physical phenomena. Of special interest in this project are equations that arise in fluid dynamics and that have application to the study of such traumatic events as hurricanes and tornadoes. The second key component of this project focuses on ergodic theory. Ergodic theory is an area of mathematics in which the subjects of probability and dynamical systems come together. It, too, has applications to real-world phenomena, predictions of weather, climate change, and earthquakes, to mention just a few. Ergodic theory usually develops in conjunction with many important parts of mathematics. It is expected that the progress in ergodic theory that will be made in this project will have an impact on our understanding of many important events in nature and find possible application in other scientific disciplines such as physics and geology.