New perspectives on dispersive equations

色散偏微分方程（PDE）模拟自然界中某些波的传播现象。他们的解决方案是随着时间的推移而在空间中传播的波，但它保存了能量。色散偏微分方程类中最著名的方程可能是非线性薛定谔方程，它模拟了各种物理波，从光纤中的信号到玻色-爱因斯坦凝聚体的宏观动力学。在过去的15年里，在解决方程解的存在性、它们的长时间行为和奇点的形成等基本问题上取得了巨大的进展。这一机构的工作的推力主要集中在确定性方面的波现象，已经研究了复杂的工具，从非线性傅立叶分析，解析数论和几何。首席研究员的研究主线一直围绕这方面的数学沿着时间。最近，越来越多的兴趣已被证明是在将非确定性的观点到色散偏微分方程领域。在过去的几年里，这已经成为主要研究者最喜欢的工作之一，也是本项目将追求的工作之一。 该项目中的另一个新元素是研究数学物理学家使用的“过程”，从一个相互作用的粒子的复杂系统到一个宏观波函数，能够描述整个系统的最重要特征。更确切地说，主要研究者将研究作为多体量子动力学的适当限制而产生的有效演化方程。从数学的角度来看，主要研究者打算在这个项目中探索的问题类型位于傅立叶分析，解析数论，数值分析，几何，概率和数学物理的交叉点。通常很难用这么多不同的数学语言进行“交流”，但现在似乎是开始这样一场对话的时候了。主要研究者在她的研究计划中提出的问题的描述应该证明这一事实。虽然她主要在纯数学领域进行研究，但她认为问题的解决方案有可能在真实的生活中看到非常具体和非常不同的结果。例如，理解通过光纤电缆发送信号的最有效方式，或者能够预测温度接近绝对零度时气体的性质，这在本质上是两种截然不同的现象。另一方面，它们都是同一色散方程（非线性薛定谔方程，这是该项目的焦点之一）的解决方案的两个方面，它们可以使用相同的数学语言进行研究。近年来，世界目睹了令人难以置信的技术进步。当科学家和工程师们寻求在技术竞技场上取得更大的进步时，他们不仅要以实验为指导，而且还要以数学模型为指导，比如对薛定谔方程的研究所提供的模型。毫无疑问，数学方面的可靠预测将有助于以最有效和最具成本效益的方式瞄准实验部分，从而节省资源和宝贵的时间。

Dispersive partial differential equations (PDE) model certain wave propagation phenomena in nature. Their solutions are waves that spread out in space as time evolves but that conserve energy. Probably the best known equation within the class of dispersive PDE is the nonlinear Schrodinger equation, which models a variety of physical waves, from signals in fiber optics to the macroscopic dynamics of the Bose-Einsten condensate. In the last fifteen years enormous progress has been made in settling fundamental questions on existence of solutions to the equation, their long-time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena that have been studied with sophisticated tools from nonlinear Fourier analysis, analytic number theory, and geometry. The principal investigator's main line of research has revolved around this aspect of mathematics for along time. More recently, a growing interest has been shown in incorporating nondeterministic points of view into the field of dispersive PDE. This has become one of the principal investigator's favorite lines of work in the last couple of years, and one that will be pursued in this project. Another new element in the project is the study of the "process" used by mathematical physicists to pass from a complex system of particles interacting with one another to a macroscopic wave function that is able to describe the most important features of the system as a whole. More precisely, the principal investigator will study the effective evolution equations arising as an appropriate limit of many body quantum dynamics. From a mathematical point of view the types of problems that the principal investigator intends to explore in this project lie at the intersection of Fourier analysis, analytic number theory, numerical analysis, geometry, probability, and mathematical physics. Often it is difficult to "communicate" in so many different mathematical languages, but the time seems ripe to start such a conversation. The description of the problems that the principal investigator proposes in her research program should testify to this fact. Although she conducts her research mostly in the realm of pure mathematics, the solutions to the problems she considers have the potential to see very concrete and very diverse consequences in real life. For example, understanding the most efficient way to send a signal through a fiber optic cable or being able to anticipate the properties of a gas when the temperature approaches absolute zero are two very different phenomena in nature. On the other hand, they are both aspects of solutions to the same dispersive equation, the nonlinear Schrodinger equation that is one of the foci of this project, and they can be studied using the same mathematical language. In recent years, the world has witnessed incredible advances in technology. As scientists and engineers seek to take even greater strides in the technological arena, they are guided by experiments, but also by mathematical models, such as those provided by research on the Schrodinger equation. There is no doubt that solid predictions on the mathematical side will help target the experimental component in the most efficient and cost-effective way possible, thus saving resources as well as valuable time.