Topological Recursion and Its Influence in Analysis, Geometry, and Topology

拓扑递归及其对分析、几何和拓扑的影响

• 批准号: 1619760
• 负责人: Motohico Mulase
• 金额: $ 4万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-05-01 至 2018-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1619760&HistoricalAwards=false
• 关键词: Topological; Recursion; Its; Influence; Analysis

This award provides partial participant support for the 2016 von Neumann Symposium, "Topological Recursion and Its Influence in Analysis, Geometry, and Topology," held July 4-8, 2016 in Charlotte, North Carolina, coordinated by the American Mathematical Society. Topological recursion is an emerging field of mathematics discovered independently in the study of random matrices and in studies of dynamics and geometry. The novelty of topological recursion is its universal applicability to many problems arising in areas of mathematics and quantum physics. For example, Catalan numbers, simple combinatorial expressions that occur in various counting problems, have generalizations associated with surfaces of complicated structure that can be calculated effectively by topological recursion. The exact same formula computes important quantities arising in the study of quantum gravity. Topological recursion not only provides theoretical formulas for many concrete problems in mathematics and physics, but also furnishes a practical computational tool. The interplay between machine computation and mathematical proof has been a driving force for recent rapid development of the field, in which many outstanding conjectures have been resolved, while new mysteries have arisen, both on the theoretical front and in computer experiments. This timely symposium, in which half of the invited speakers are early-career researchers, is aimed to further advance this exciting, rapidly-developing field.Topological recursion is a new emerging field of mathematics developed in statistical mechanical study of random matrices. Independently, essentially the same structure of the theory was discovered in research on the volume of moduli spaces of bordered hyperbolic surfaces. Due to the simple nature of concrete recursive formulas, topological recursion relates many current research frontiers of mathematics in a novel and understandable way. For example, counting Hurwitz numbers, Gromov-Witten theory, Gaiotto's conjecture on opers arising from string theory, the WKB analysis of Schroedinger equations, and the relation between A-polynomials and colored Jones polynomials for knots and their generalizations are all deeply related through ideas stemming from topological recursion. The goal of the symposium is to promote interest among young researchers in this exciting research frontier and to significantly enhance the subject matter by disseminating recent progress and identifying important problems for future development. The conference website is www.ams.org/meetings/amsconf/symposia/symposia-2016

该奖项为2016年冯·诺伊曼研讨会“拓扑递归及其对分析、几何和拓扑的影响”提供了部分参与者支持，该研讨会于2016年7月4日至8日在北卡罗来纳州夏洛特举行，由美国数学学会协调。拓扑递归是一个新兴的数学领域，在随机矩阵的研究以及动力学和几何的研究中被独立发现。拓扑递归的新颖之处在于它对数学和量子物理领域中出现的许多问题的普遍适用性。例如，加泰罗尼亚数，出现在各种计数问题中的简单组合表达式，具有与复杂结构曲面相关的泛化，可以通过拓扑递归有效地计算。同样的公式也可以计算量子引力研究中出现的重要量。拓扑递归不仅为许多具体的数学和物理问题提供了理论公式，而且提供了一种实用的计算工具。机器计算和数学证明之间的相互作用是近年来该领域快速发展的推动力，在理论和计算机实验方面，许多杰出的猜想已经得到解决，同时新的谜团也出现了。这个及时的研讨会，其中一半的受邀演讲者是早期的职业研究人员，旨在进一步推进这一令人兴奋的，快速发展的领域。拓扑递归是在随机矩阵的统计力学研究中发展起来的一个新兴数学领域。独立地，在对有边界双曲曲面的模空间体积的研究中发现了本质上相同的理论结构。由于具体递归公式的简单性，拓扑递归以一种新颖易懂的方式将当前数学的许多研究前沿联系起来。例如，计算Hurwitz数、Gromov-Witten理论、Gaiotto关于弦论中粒子的猜想、WKB对薛定谔方程的分析、结的a多项式与彩色琼斯多项式的关系及其推广等，都是通过拓扑递归的思想产生的。研讨会的目标是促进年轻研究人员对这一令人兴奋的研究前沿的兴趣，并通过传播最近的进展和确定未来发展的重要问题来显著加强这一主题。会议网站是www.ams.org/meetings/amsconf/symposia/symposia-2016