Symplectic and Poisson Geometry in interaction with Algebra, Analysis and Topology

辛几何和泊松几何与代数、分析和拓扑的相互作用

• 批准号: 0965738
• 负责人: Maciej Zworski
• 金额: $ 3.8万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-03-01 至 2011-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0965738&HistoricalAwards=false
• 关键词: Symplectic; Poisson; Geometry; interaction; Algebra

This award will provide funding to organize a conference,``Symplectic and Poisson geometry in interaction with Algebra, Analysis and Topology'', celebrating four decades since the emergence of symplectic and Poisson geometry and their influence on major areas of mathematics. The conference focuses on recent important developments in symplectic and Poisson geometry, and the interactions of these fields with Analysis, Algebra, differential equations and low-dimensional topology. Specific topics covered by the talks will include: Taubes' recent proof of the Weinstein conjecture using Seiberg-Witten theory, recent progress in Lagrangian intersection theory, classical and quantum Yang-Baxter equations, Poisson and quantum groupoids, dynamical Weyl groups, q-deformed Casimir connections and Kazdhan-Lusztig functors. The conference will provide a forum to outline the recently found connections by Nicolai Reshetikhin, San Vu-Ngoc and others between integrable systems in symplectic and algebraic geometry and representation theory. Reshetikhin and Vu-Ngoc talks will also discuss the recent progress in the quantization of integrable systems from a more algebraic and a more geometric view point, respectively. Other topics covered in the conference will regard recent breakthroughs in relating geodesic flow to eigenfunctions, and Hitrik and Sjostrand's recent work on spectra of non-self adjoint operators in dimension two (which relies heavily on Alan Weinstein's famous work on spectra of Zoll surfaces). The talks by Tudor Ratiu and Jerrold Marsden will focus on applications of symplectic geometry to a wide problems in physics and engineering such as as fluid and plasma theory, liquid crystals and micropolar fluids.The goal behind this conference is that of holding a high profile meeting to bring together world experts and junior researchers to discuss these current exciting interactions. The time of the conference (May 2010) coincides with the first year anniversary of Alan Weinstein?s retirement from UC Berkeley. Weinstein has been one of the most influential figures in symplectic geometry and analysis in the past forty years. His fundamental work has inspired many mathematicians and led to the development of central concepts in symplectic and Poisson geometry, as well as to the establishment of symplectic geometry as an independent discipline within mathematics. The conference will provide a forum to dicuss Weinstein's impact on geometry and mathematics at large. The last few decades have witnessed numerous spectacular interactions between symplectic geometry, analysis, low dimensional topology and partial differential equations leading to new understanding in fundamental problems of mathematics. Today symplectic geometry is an active, central branch of mathematics populated by deep results and connections with physics, low-dimensional topology, gauge theory, integrable systems, representation theory, group theory, semiclassical analysis and Lie groups. The main theme of the Conference is to illuminate the particular type of interactions which characterize the past forty years of developments in symplectic geometry. To this end the conference will have talks by leading experts, both junior and senior, describing the current state of the art of several of the most fundamental research problems in these areas. Symplectic and Poisson geometry are by now well established fields of research, and its language and techniques are being used in many areas of mathematics, theoretical physics, and engineering such as symmetric bifurcation problems, integrable systems, string theory, geometric phases, nonlinear control, nonholonomic mechanics and locomotion generation in robotics.

该奖项将提供资金组织一次“辛几何和泊松几何与代数、分析和拓扑相互作用”的会议，庆祝辛几何和泊松几何出现四十周年及其对数学主要领域的影响。会议重点讨论辛几何和泊松几何的最新重要发展，以及这些领域与分析、代数、微分方程和低维拓扑的相互作用。会谈涵盖的具体主题包括：Taubes 最近使用 Seiberg-Witten 理论证明韦恩斯坦猜想、拉格朗日交集理论的最新进展、经典和量子 Yang-Baxter 方程、泊松和量子群群、动力学 Weyl 群、q 变形卡西米尔连接和 Kazdhan-Lusztig 函子。会议将提供一个论坛，概述 Nicolai Reshetikhin、San Vu-Ngoc 等人最近发现的辛几何和代数几何与表示论中可积系统之间的联系。 Reshetikhin 和 Vu-Ngoc 的演讲还将分别从更代数和更几何的角度讨论可积系统量化的最新进展。会议涵盖的其他主题包括最近在测地线流与本征函数相关方面的突破，以及 Hitrik 和 Sjostrand 最近在二维非自伴随算子谱方面的工作（这在很大程度上依赖于 Alan Weinstein 关于 Zoll 曲面谱的著名工作）。 Tudor Ratiu 和 Jerrold Marsden 的演讲将重点讨论辛几何在物理和工程领域广泛问题中的应用，例如流体和等离子体理论、液晶和微极性流体。本次会议的目标是举办一次高调会议汇集世界各地的专家和初级研究人员，讨论当前令人兴奋的互动。会议召开的时间（2010年5月）恰逢艾伦·韦恩斯坦从加州大学伯克利分校退休一周年。韦恩斯坦是过去四十年来辛几何和分析领域最有影响力的人物之一。他的基础工作启发了许多数学家，并导致辛几何和泊松几何中心概念的发展，以及辛几何作为数学中独立学科的建立。该会议将提供一个论坛来讨论韦恩斯坦对几何和数学的影响。在过去的几十年里，辛几何、分析、低维拓扑和偏微分方程之间发生了许多引人注目的相互作用，导致了对数学基本问题的新理解。今天，辛几何是数学的一个活跃的中心分支，充满了深刻的结果以及与物理学、低维拓扑、规范理论、可积系统、表示论、群论、半经典分析和李群的联系。会议的主题是阐明过去四十年辛几何发展特征的特殊类型的相互作用。 为此，会议将由初级和高级专家进行演讲，描述这些领域中几个最基本的研究问题的当前状态。辛几何和泊松几何现已成为成熟的研究领域，其语言和技术被用于数学、理论物理和工程的许多领域，例如对称分岔问题、可积系统、弦理论、几何相位、非线性控制、机器人技术中的非完整力学和运动生成。