SM: Geometry and Topology of Moduli Spaces and Applications

多年来，理解黎曼曲面的模空间的几何和拓扑以及相应的映射类群一直是数学中一个重要的目标。在过去的15年中，出现了一些关于模空间的新观点，这些观点不仅增加了我们对这些重要对象的理解，而且从根本上影响了拓扑和几何的几个领域的主要研究方向，包括双曲几何和几何群论，代数和辛几何，以及最近的代数拓扑。在过去的五年中，在这些领域中有几个领域取得了惊人的进展。作为一个整体，这些领域在过去几年中代表了拓扑学和几何学中一些最令人兴奋的研究方向，并且它们承诺在可预见的未来继续这样做。此提案是为模空间的拓扑和几何及相关主题的一个主要的，为期三年的重点项目提供资金。每年都将强调这个主题的不同数学视角，但所有三年都将涉及来自广泛子领域的参与者。三个重点领域将是双曲几何和几何群论，模空间的代数拓扑和弦拓扑，以及模空间的代数几何和辛几何。自19世纪中期黎曼时代以来，对曲面的研究一直是数学的主要推动力。在给定的二维曲面上的几何结构空间称为黎曼曲面的模空间。这些模空间在代数几何中得到了经典的研究。随着20世纪70年代M. Gromov的开创性工作，这些模空间也成为辛几何研究的工具。它们也是瑟斯顿在同一时期提出的低维拓扑和双曲几何的现代观点的核心。随着20世纪80年代共形场论和弦理论的发展，这些模空间也开始在理论物理中发挥重要作用。最近几年，代数拓扑技术被引入到这些模空间的研究中，并取得了令人兴奋的结果。相反，来自物理和几何的形式化对代数拓扑的最新研究方向产生了重大影响。在过去的五年中，在所有这些影响和影响黎曼曲面模空间的几何和拓扑领域都取得了令人兴奋的进展。可以想象，在这些研究领域产生的兴奋吸引了许多研究生和年轻的数学家。要成为有效的研究人员，重要的是这些年轻的数学家获得对这些模空间和相关对象的各种不同观点的理解。这些领域在技术和研究方向上的交叉授粉，可以对拓扑和几何这些中心主题的发展产生强大的影响。此提案是为模空间的拓扑和几何及相关主题的一个主要的，为期三年的重点项目提供资金。该项目将由斯坦福大学数学研究中心和美国数学研究所共同组织。斯坦福大学数学研究中心是几何和拓扑研究的主要中心之一，美国数学研究所是一个主要的独立研究机构。每年都将强调这个主题的不同数学视角，但所有三年都将涉及来自广泛子领域的参与者。一些世界领先的高级数学家，他们的初级同事，以及学生将参加这些项目，分享和比较他们不同的观点和专业领域，并将共同努力加深我们对这一数学中心领域的理解，并产生新的和令人兴奋的方法，技术和结果。

Understanding the geometry and topology of the moduli space of Riemann surfaces and the corresponding mapping class groups has been a goal of central importance in mathematics for many years. In the last 15 years there have been several new perspectives on moduli spaces that have not only increased our understanding of these important objects, but have fundamentally affected major research directions of several areas of topology and geometry, including Hyperbolic Geometry and Geometric Group Theory, Algebraic and Symplectic Geometry, and most recently, Algebraic Topology. In the last five years there have been several startling advances in several of these areas. Taken as a whole, these areas have, in the last few years, represented some of the most exciting directions of study in topology and geometry, and they promise to continue to do so in the forseeable future. This proposal is for the funding of a major, three year emphasis program in the topology and geometry of moduli spaces and related topics. Each year a different mathematical perspective of this topic will be emphasized, but all three years will involve participants from a broad range of subfields. The three areas of emphasis will be Hyperbolic Geometry and Geometric Group Theory, The Algebraic Topology of Moduli Spaces and String Topology, and The Algebraic Geometry of Moduli Spaces and Symplectic Geometry.The study of surfaces has been a major driving force in mathematics since the time of Riemann in the mid 19th century. The space of geometric structures on a given two dimensional surface is known as the moduli space of Riemann surfaces. These moduli spaces have been classically studied in algebraic geometry. With the pioneering work of M. Gromov in the 1970's, these moduli spaces became instrumental in the study of symplectic geometry as well. They are also central in the modern view of low dimensional topology and hyperbolic geometry initiated by Thurston around the same time. With the development of conformal field theory and string theory in the 1980's, these moduli spaces also began to play an important role in theoretical physics. Most recently, techniques of algebraic topology have been brought to bear on the study of these moduli spaces over the last few years with exciting results. Conversely, formalisms from physics and geometry have had a major impact on recent research directions in algebraic topology. The last five years have seen exciting developments in all these geometric and topological areas affecting and affected by moduli spaces of Riemann surfaces. As one can imagine, the excitement produced in these areas of study have attracted many graduate students and young mathematicians. To be effective researchers, it is important that these young mathematicians gain an understanding of the various different perspectives on these moduli spaces and related objects. Cross pollination between these areas both in terms of techniques and directions of research, can have a powerful effect on the development of these central topics in topology and geometry. This proposal is for the funding of a major, three year emphasis program in the topology and geometry of moduli spaces and related topics. The program will be organized by the Mathematics Research Center of Stanford University, one of the leading centers of research in geometry and topology, and by the American Institute of Mathematics, which is a major independent research institute. Each year a different mathematical perspective of this topic will be emphasized, but all three years will involve participants from a broad range of subfields. Some of the world's leading senior mathematicians, their junior colleagues, as well as students will participate in these programs, share and compare their different perspectives and areas of expertise, and will work together to deepen our understanding of this central area of mathematics, and produce new and exciting methods, techniques, and results.