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

University Of California-Davis, Davis CA

Investigators

Abstract

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