Conference on Algebraic Geometry, Mathematical Physics, and Solitons

该项目将于2022年10月7日至9日在哥伦比亚大学举办为期三天的会议，主题为“代数几何、数学物理和孤子”。该活动将汇集来自各个领域的早期职业研究人员和世界知名数学家，包括代数几何和数学物理。会议将为研究生和博士后提供一个相互交流的机会，并与来自广泛数学学科的数学家交流。来自一些领先专家的广泛观点对早期职业研究人员来说是非常宝贵的。可积系统最初是在物理学中研究的，目的是为了描述经典力学中出现的运动积分。可积系统的思想和方法在数学的许多领域产生了广泛的影响。例如，肖特基问题是黎曼曲面理论中的一个经典问题，它与各种可积层次，Korteweg-De Vries和Kadomtsev-Petviashvili方程的研究密切相关。Korteweg-De Vries方程是一个非线性偏微分方程，它以具有非线性波的孤子解而著称。最近，Korteweg-De Vries方程出现在关于g格紧化黎曼曲面模空间的枚举几何的Witten猜想中。这些猜想由Kontsevich， Okounkov-Pandharipande和Mirzakhani解决，并导致了与动力系统，双曲几何和Gromov-Witten理论的惊人联系。可积系统的其他方面与对称群和更一般的代数结构（如量子群、环群和Cherednik代数）有关。在原始物理动机方面，可积系统控制着各种超对称量子场论和随机矩阵的行为。这次会议的目标是把这些不同的数学和数学物理的分支集合在一起。会议网站可在https://math.columbia.edu/~agmps22/.This上找到奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will conduct a three day conference on the topic of "Algebraic Geometry, Mathematical Physics, and Solitons", to be hosted at Columbia University from October 7-9, 2022. This event will bring together early career researchers and world-renowned mathematicians from a variety of fields, including algebraic geometry and mathematical physics. The conference will be an opportunity for graduate students and postdoctoral fellows to interact with each other and with mathematicians from a wide range of mathematical disciplines. This broad perspective from some of the leading experts will be invaluable to early career researchers.Integrable systems were originally studied in physics in order to describe the integrals of motion arising in classical mechanics. The ideas and methods of integrable systems have since had a wide influence in many areas of mathematics. For example, the Schottky problem, a classical problem in the theory of Riemann surfaces, is deeply connected to the study of various integrable hierarchies, the Korteweg-De Vries and Kadomtsev-Petviashvili equations. The Korteweg-De Vries equation, a nonlinear partial differential equation, is notable for having soliton solutions, which are a type of nonlinear wave. More recently, the Korteweg-De Vries equation has appeared in the Witten conjectures on the enumerative geometry of the moduli space of compact Riemann surfaces of genus g. These conjectures were solved by Kontsevich, Okounkov-Pandharipande, and Mirzakhani, and led to surprising connections with dynamical systems, hyperbolic geometry, and Gromov-Witten theory. Other aspects of integrable systems connect to symmetry groups and to more general algebraic structures such as quantum groups, loop groups, and Cherednik algebras. In terms of the original physical motivation, integrable systems control the behavior of various supersymmetric quantum field theories and random matrices. The goal of the conference is to bring together these many different strands of mathematics and mathematical physics. The conference website can be found at https://math.columbia.edu/~agmps22/.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.