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个名为“辛和泊松几何与代数、分析和拓扑的相互作用”的会议，庆祝辛和泊松几何出现40周年及其对数学主要领域的影响。会议的重点是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。讲座的具体主题包括：Taubes最近用Seiberg-Witten理论证明了Weinstein猜想，拉格朗日交点理论的最新进展，经典和量子Yang-Baxter方程，泊松和量子群，动态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等人最近发现的可积系统在辛几何和代数几何以及表征理论之间的联系。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。会议中涉及的其他主题将包括最近在将测地线流与特征函数联系起来方面的突破，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的讲座将重点关注辛几何在物理和工程领域的广泛应用，如流体和等离子体理论、液晶和微极流体。这次会议的目的是举办一次高知名度的会议，汇集世界专家和初级研究人员，讨论这些当前令人兴奋的相互作用。会议召开的时间（2010年5月）正好是艾伦·韦恩斯坦（Alan Weinstein）的一周年纪念。他从加州大学伯克利分校退休。温斯坦是近四十年来辛几何和辛分析领域最具影响力的人物之一。他的基本工作启发了许多数学家，并导致了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一个独立学科的建立。会议将提供一个论坛来讨论温斯坦对几何和数学的影响。过去几十年见证了辛几何、分析、低维拓扑和偏微分方程之间无数壮观的相互作用，导致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，它与物理学、低维拓扑、规范论、可积系统、表示论、群论、半经典分析和李群有着深刻的结果和联系。会议的主题是阐明过去四十年来辛几何发展中所特有的特殊类型的相互作用。为此，会议将由高级和初级的主要专家进行会谈，描述这些领域中几个最基本的研究问题的当前状态。辛几何和泊松几何是一个成熟的研究领域，其语言和技术被应用于数学、理论物理和工程的许多领域，如对称分岔问题、可积系统、弦理论、几何相、非线性控制、非完整力学和机器人运动生成。