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

该奖项为2016年冯·诺依曼研讨会提供部分参与者支持，“拓扑递归及其在分析，几何和拓扑中的影响”，于2016年7月4日至8日在北卡罗来纳州的夏洛特举行，由美国数学学会协调。拓扑递归是一个新兴的数学领域，在随机矩阵的研究中独立发现，并在动力学和几何学的研究。拓扑递归的新奇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，卡塔兰数（Catalan numbers），一种出现在各种计数问题中的简单组合表达式，具有与复杂结构的表面相关的推广，可以通过拓扑递归有效地计算。在量子引力的研究中，同样的公式也计算出了重要的物理量。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，而且也是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一直是该领域最近快速发展的驱动力，其中许多突出的问题已经解决，同时在理论前沿和计算机实验中出现了新的谜团。 本次研讨会的目的是进一步推动这一令人兴奋的快速发展的领域。拓扑递归是随机矩阵统计力学研究中发展起来的一个新兴数学领域。独立地，基本上相同的结构的理论被发现在研究的体积模空间的有边双曲曲面。由于具体递归公式的简单性质，拓扑递归以一种新颖和可理解的方式联系了许多当前数学的研究前沿。例如，Hurwitz数的计数、Gromov-Witten理论、Gaiotto的弦论猜想、Schroedinger方程的WKB分析、A-多项式和色琼斯多项式之间的关系以及它们的推广都与拓扑递归的思想密切相关。研讨会的目标是促进青年研究人员对这一令人兴奋的研究前沿的兴趣，并通过传播最新进展和确定未来发展的重要问题来大大加强这一主题。会议网址为www.ams.org/meetings/amsconf/symposia/symposia-2016

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