Program in Nonlinear Waves, Kinetic Theory and Hamiltonian Partial Differential Equations-Fields Institute, Spg 04

摘要奖：DMS-0352061主要研究者：Nicholas M. Ercolani这一建议是为了支持美国的初级数学家参加科学活动领域研究所在专题计划学期在非线性波，动力学理论和汉密尔顿偏微分方程（PDE），将在2004年春季学期。本学期的重点将放在偏微分方程的研究领域，这些研究领域是由非线性波动理论、动力学理论和哈密顿系统所激发的。Hamilton偏微分方程是一类线性和非线性的偏微分方程，它们都具有一个共同的性质，即它们可以用各种形式的辛结构写成具有无穷多个自由度的Hamilton系统的形式。主要的例子包括非线性波动方程、非线性薛定谔方程和水波的欧拉方程。与动力系统的类比提出了一些基本问题，这些问题构成了本学期重点讨论哈密尔顿偏微分方程的部分动机。特别值得注意的是，已经建立的联系之间的理论的动力学方程来自criticalmechanics，和非线性系统的偏微分方程，出现在他们的宏观限制。该范例是玻尔兹曼方程的流体动力学标度极限，但也有许多新兴领域的相关分析，包括一个数学上的非线性波湍流理论的严格基础的开端。组织者希望这个主题计划是一个非常动态的研究重点偏微分方程模型的各种物理现象，如行星运动，激光和光纤系统，海浪和流体湍流。该计划的特点是广泛的国际性，它代表了一个机会，暴露为年轻的数学家。短期课程系列，四个讲习班和专题讨论会是特别适合参与发展研究数学家或物理学家在他们的职业生涯早期。组委会相信，该项目将使来自不同学科的参与者之间产生进一步的互动和持久的合作。此外，将积极鼓励妇女和代表性不足的少数民族的参与。

AbstractAward: DMS-0352061Principal Investigator: Nicholas M. ErcolaniThis proposal is for support of junior US based mathematicians toparticipate in scientific activities at the Fields Instituteduring the thematic program semester in Nonlinear Waves, KineticTheory and Hamiltonian Partial Differential Equations (PDE) thatwill take place during the Spring semester of 2004. The focus ofthe semester will concern areas of research on PDE that aremotivated by nonlinear wave theory, kinetic theory, andHamiltonian systems. Hamiltonian PDE form a class of linear andnonlinear partial differential equations which share the propertythat they can be written in the form of a Hamiltonian system withinfinitely many degrees of freedom, using various and sometimesnonclassical symplectic structures. Principal examples includenonlinear wave equations, nonlinear Schroedinger equations andEuler's equations for water waves. The analogy with dynamicalsystems raises a number of basic questions, which form a part ofthe motivation for this semester of focus on Hamiltonian PDE. Ofparticular note is the connection that has been establishedbetween the theory of kinetic equations coming from statisticalmechanics, and the nonlinear systems of PDE which arise in theirmacroscopic limits. The paradigm is the fluid dynamical scalinglimit of the Boltzmann equation, but there are numerous emergingareas of relevance for this analysis, including the beginnings ofa mathematically rigorous foundation for the theory of nonlinearwave turbulence.The organizers are expecting this thematic program to be a verydynamic focus of research on Partial Differential Equations thatmodel a variety of physical phenomena, such as planetary motions,lasers and optical fiber systems, ocean waves and fluidturbulence. The character of the program is broadlyinternational, and it represents an opportunity for exposure foryoung mathematicians. The short course series, the four workshopsand the symposia are especially appropriate for participation bydeveloping research mathematicians or physicists early in theircareer. The organizing committee believes that the program willengender further interaction and lasting collaboration amongparticipants from various disciplines. Additionally, theparticipation of women and under-represented minorities will beactively encouraged.