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

摘要奖：DMS-0352061首席研究员：Nicholas M.Ercolani这项建议是为了支持美国初级数学家在2004年春季学期的主题项目学期中参与菲尔德研究所的科学活动，主题项目是非线性波动、运动论和哈密尔顿偏微分方程(PDE)。本学期的重点将涉及由非线性波动理论、动力学理论和哈密顿系统推动的偏微分方程组的研究领域。哈密顿偏微分方程组形成了一类线性和非线性偏微分方程组，它们的共同性质是它们可以写成无限多个自由度的哈密顿系统的形式，使用各种有时非经典的辛结构。主要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。与动力学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学习哈密顿偏微分方程的动机的一部分。特别值得注意的是，来自统计力学的动力学方程理论与出现在其宏观极限内的偏微分方程组的非线性系统之间已经建立了联系。这个范例是玻尔兹曼方程的流体动力学标度极限，但这种分析有许多新的相关领域，包括开始为非线性波浪湍流理论奠定严格的数学基础。组织者希望这个主题项目成为偏微分方程组研究的一个非常动态的焦点，它模拟了各种物理现象，如行星运动、激光和光纤系统、海浪和流体湍流。该项目的特点是广泛的国际化，它为年轻的数学家提供了一个展示自己的机会。短期课程系列、四个工作坊和专题讨论会特别适合于在研究数学家或物理学家的职业生涯早期发展他们的参与。组委会相信，该项目将在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，还将积极鼓励妇女和任职人数不足的少数群体参加。

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.