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

近年来，自然二维(2D)数学结构的概率理论取得了重大进展，包括度量、度量、函数和曲面上的曲线。一个被称为Liouville量子重力(LQG)表面的单参数随机表面家族已经成为研究这种结构的一个富有成效的环境。LQG起源于理论物理中对二维量子引力和弦理论的研究。随后，LQG成为概率论中一门活跃而深入的数学学科。由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，特别是对来自代表性不足群体的学生、来自地理弱势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学对应。在物理学中，二维量子引力与共形物质的耦合由LQG描述，该理论由被称为Liouville共形场理论的共形场理论(CFT)所支配。2D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的随机平面映射模型在比例极限下收敛到LQG，为这一物理图像奠定数学基础。最具挑战性的两种情况是随机几何体不均匀或底层曲面非简单连接。该项目旨在解决这两种情况下的公开问题。除了几何方面，Liouville CFT的关联函数还具有深代数结构，这些结构有望通过一个称为共形自举的示意性程序来计算。研究的另一个主要目标是严格建立Liouville CFT的共形自举程序。为了实现这些目标，PI将依靠LQG和一系列称为Schramm-Loewner进化的随机平面曲线之间的丰富相互作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.