Special meeting: Dynamical systems and evolution equations, CRM

该学院2008年冬季学期的主题课程的重点是动力系统，从广义上解释，包括在微分几何和数学物理中的基本问题的应用。讨论的主题包括：(1)动力系统和偏微分方程之间的相互作用，特别是在哈密顿系统的背景下，(2)几何发展方程，如Ricci流和非本征曲率流，(3)谱理论及其与哈密顿动力学的关系，以及(4)Floer理论和哈密顿流。在过去几年中，在这四个领域取得了巨大成就，在这一领域的一些最基本和最困难的问题上取得了进展。这些进展对几何学和拓扑学的最新进展产生了广泛的影响，也揭示了基本的物理过程，如由常微分方程和偏微分方程式模拟的非线性波动现象。本项目学期的目的是将对这些主题感兴趣的不同国际研究社区的成员聚集在一起，就相关主题提供一系列高级课程，以便使该领域的新研究人员能够接触到该主题，并讨论该领域下一步进展和方向的观点和一般指示。2008年冬季CRM的主题学期的中心焦点是动力学系统。动力系统理论是关于系统随时间演化的描述。这样的系统是基本的，在物理、化学和生物现象的建模中以及在几何和许多其他数学领域中非常常见。在过去的几年里，动力系统领域取得了巨大的成就，包括G.Perelman的Poincaré猜想的证明(这一事件通过2006年菲尔兹奖牌颁奖的戏剧性事件而为公众所知)。这些进展对几何和拓扑学的最新进展产生了广泛的影响。现代动力系统理论也是研究许多基本物理过程的基础，并用常微分方程式和偏微分方程式进行建模。本项目学期的目的是将对这些主题感兴趣的不同国际研究人员社区的代表聚集在一起。其活动将包括(1)一系列关于该主题的高级课程，以便向学生和新的研究人员提供该领域，(2)主持关于该领域下一步进展和进展领域的观点和未来方向的讨论。

The focus of the thematic program semester of winter 2008 at the CRM is on dynamical systems, interpreted in a broad sense so as to include applications to fundamental problems in differential geometry as well as in mathematical physics. Topics that are considered include:(1) the interplay between dynamical systems and PDE, in particular in the context of Hamiltonian systems,(2) geometric evolution equations such as Ricci flows and extrinsic curvature flows, (3) spectral theory and its relationship to Hamiltonian dynamics, and(4) Floer theory and Hamiltonian flows. In the past several years there have been dramatic achievements in these four areas, representing progress on a number of the most basic and difficult questions in this field. These advances have had a broad impact on recent progress in geometry and topology, and they also shed light on basic physical processes, such as nonlinear wave phenomena, that are modeled by ordinary and partial differential equations.The purpose of this program semester is to bring together members of the diverse international community of researchers who have an interest in these topics, to give a series of advanced-level courses on relevant subject matter so as to make the topic accessible to new researchers in the field, and to bring into discuss the perspectives and general indications for the next advances and directions of progress in the area.The central focus of the theme semester of winter 2008 at CRM is dynamical systems. The theory of dynamical systems is concerned with the description of the evolution of systems depending on time. Such systems are fundamental, and appear very commonly in the modeling of physical, chemical and biological phenomena, as well as in geometry and many other areas of mathematics. In the past several years there have been dramatic achievements in the area of dynamical systems, including the proof of Poincaré conjecture by G. Perelman (an event known to the public through the drama of the Fields medal awards in 2006). These advances have had a broad impact on recent progress in geometry and topology. The modern theory of dynamical systems has also been fundamental in the study of many basic physical processes, and their modeling by ordinary and partial differential equations. The purpose of this program semester is to bring together representatives of the diverse international community of researchers who have an interest in these topics. Its activities will comprise (1) a series of advanced-level courses on the subject matter, so as to make the field available to students and new researchers, (2) to host discussions of the perspectives and future directions for the next advances and areas of progress in the field.