CAREER: Liouville Quantum Gravity, Two-Dimensional Random Geometry, and Conformal Field Theory

最近在自然二维（2D）数学结构的概率论方面取得了重大进展，包括度量、测度、函数和曲面上的曲线。一个单参数的随机表面，称为刘维尔量子引力（LQG）表面，已经成为一个富有成效的设置来研究这样的结构。LQG起源于理论物理学中的二维量子引力和弦理论的研究。随后，LQG成为概率论中一个活跃而深刻的数学课题。该奖项资助的项目的研究部分旨在解决LQG数学理论中的突出挑战，从而为理论物理中的几个假设提供坚实的基础。该项目的教育部分旨在使高质量的数学教育和研究更容易获得，特别是来自代表性不足群体的学生，地理条件不利地区的研究人员和高中数学教师。量子引力是随机几何的物理对应物。在物理学中，认为与共形物质耦合的二维量子引力是由LQG描述的，由称为Liouville CFT的共形场论（CFT）支配。二维量子引力最直观的表述是通过其微观描述，即随机平面映射。该项目的主要研究目标是证明某些经典随机平面映射模型在标度极限下收敛于LQG，为这一物理图景奠定数学基础。两种最具挑战性的情况是当随机几何形状是不均匀的或当底层表面是非单连通的。该项目旨在解决这两种情况下的未决问题。除了几何方面，刘维CFT的相关函数具有深刻的代数结构，预计将计算一个示意性的程序称为共形引导。研究的另一个主要目标是严格建立Liouville CFT的共形引导程序。为了实现这些目标，PI将依靠LQG和一系列随机平面曲线之间的丰富相互作用，称为Schramm-Loewner evolution。该奖项反映了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。

Significant recent advances have been made in the probability theory of natural two-dimensional (2D) mathematical structures, including metrics, measures, functions, and curves on surfaces. A one-parameter family of random surfaces, called Liouville quantum gravity (LQG) surfaces, has emerged as a fruitful setting to study such structures. LQG originated from the study of 2D quantum gravity and string theory in theoretical physics. Subsequently, LQG has become an active and deep mathematical subject in probability. The research part of the project funded by this award aims to address outstanding challenges in the mathematical theory of LQG, thereby providing a firm foundation for several assumptions in theoretical physics. The educational part of the project aims to make high quality mathematical education and research more accessible, in particular, to students from underrepresented groups, researchers from geographically disadvantaged areas, and high school mathematics teachers.Quantum gravity is the physics counterpart of random geometry. It is believed in physics that 2D quantum gravity coupled with conformal matter is described by LQG, governed by a conformal field theory (CFT) called Liouville CFT. The most intuitive formulation of 2D quantum gravity is through its microscopic description, namely random planar maps. The primary research goal of the project is to show that certain classical random planar map models converge to LQG in the scaling limit, laying a mathematical foundation for this physical picture. The two most challenging cases are when the random geometry is non-uniform or when the underlying surface is non-simply-connected. The project aims to address open questions in both cases. Besides the geometric aspect, the correlation functions of Liouville CFT possess deep algebraic structures, which are expected to be computed by a schematic program called the conformal bootstrap. Another major goal of the research is to rigorously establish the conformal bootstrap program for Liouville CFT. To achieve these goals, the PI will rely on the rich interplay between LQG and a family of random planar curves called the Schramm-Loewner evolution.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.