Conference Proposal: Unity in Mathematics

主要研究人员：Pavel Etingof，Joseph Harris，Isadore Singer建议编号：DMS-0315184机构：麻省理工学院标题：数学统一摘要：数学统一会议的目标是讨论数学的重要最新发展，绘制有希望的新方向，并关注不同领域之间的联系。具体地说，会议将强调近年来取得重大进展的两个领域，即1)几何和物理，2)表示理论。几何学和物理学的进展的讨论将集中在来自量子场论和弦理论的新思想，以及它们的几何后果。这一讨论将涉及不同的主题，如镜像对称性，Seiberg-Witten理论，扭曲K-理论，手征代数，非对易Yang-Mills理论，形式理论，模空间上的交集理论等。表示理论的进展将集中在表示理论与几何之间的相互作用；这些相互作用最近导致了强大的新的表示研究方法的发展。其中一些方法包括几何朗兰兹程序、箭图变种、量子群的几何实现及其表示。“数学的统一”会议的目的是聚集不同的资深数学家和青年研究人员，讨论数学领域最重要的最新发展，重点是“几何和物理”以及“表示论”。第一个主题涉及几何学的发展，这些发展是由量子场论(描述基本粒子的运动)和弦理论(试图统一量子场论和爱因斯坦的广义相对论)最近的突破引发的。尽管弦理论还没有做出成功的实验预测，但令人惊讶的是，它确实在几何学上做出了成功的预测。这些预测是科学史上第一个将几何学作为理论物理的“实验室”的例子，并提供了更多的证据，证明弦理论是“正确的”，就像实验验证一样。第二个主题涉及表示理论(可以非常粗略地定义为对称谱的代数的一个分支)的发展，这是由于系统地使用几何和拓扑学(形状理论)中的深层思想而产生的。这次会议将吸引许多在不同领域工作的数学家，以及许多理论物理学家。预计年轻的研究人员将发现这次会议特别有用，因为它将讨论在领域之间的边界发生的发展。对于那些接受过狭隘专业培训的人来说，这很可能是一个顿悟。这次会议将是一个独特的场合；杰出的数学家们不仅将回顾最近的进展，还将讨论它们对数学的广泛影响。将制定特别规定，鼓励年轻研究人员、研究生、妇女和少数族裔参与。

Principal Investigator: Pavel Etingof, Joseph Harris, Isadore SingerProposal Number: DMS-0315184Institution: Massachusetts Institute of Technology Title: Unity in MathematicsAbstract:The goals of the conference "Unity of mathematics" are to discuss important recent developments in mathematics, to chart promising new directions, and to focus on connections between different fields. Specifically, the conference will emphasize two areas where there have been significant advances in recent years, namely 1) geometry and physics, and 2) representation theory. The discussion of advances in geometry and physics will focus on new ideas coming from quantum field theory and string theory, and their geometric consequences. This discussion will involve diverse topics, such as mirror symmetry, Seiberg-Witten theory, twisted K-theory, chiral algebras, noncommutative Yang-Mills theory, formality theory, intersection theory on moduli spaces, etc. The discussion of advances in representation theory will focus on the interactions between representation theory and geometry; these interactions have recently led to the development of powerful new methods of studying representations. Some of the methods are the geometric Langlands program, quiver varieties, geometric realizations of quantum groups and their representations. The conference ``Unity of Mathematics'' is designed to bring together a diverse group of leading senior mathematicians and young researchers for a discussion of the most important recent developments in mathematics, with the focus on the topics "geometry and physics", and "representation theory". The first topic involves developments in geometry that were triggered by recent breakthroughs in quantum field theory (which describes the motion of elementary particles) and string theory (which attempts to unify quantum field theory and Einstein's general relativity). Although string theory has not yet made successful experimental predictions, it is striking that it did make successful predictions in geometry. These predictions are the first example in the history of science when geometry served as a ``lab'' for theoretical physics, and provide as much evidence that string theory is ``correct'' as an experimental verification would be. The second topic involves developments in representation theory (a branch of algebra that could be very roughly defined as spectroscopy of symmetry), which have resulted from systematic use of deep ideas from geometry and topology (theory of shape). The conference will attract numerous mathematicians working in different fields, as well as many theoretical physicists. Young researchers are expected to find the conference especially useful because it will discuss developments occurring at boundaries between fields. This may well be an epiphany to those trained in a narrow specialty. The meeting will be a unique occasion; prominent mathematicians will not only review recent advances but will discuss their broad implications for mathematics. Special provisions will be made to encourage young researchers, graduate students, women and minorities to participate.