Liouville Quantum Gravity and Its Applications

这个项目将研究刘维尔量子引力(LQG)，这是一种关于随机表面的理论。这些随机表面出现在理论物理的各个领域。在玻色弦理论中，它们可以用来模拟弦在空间中移动所扫过的表面。它们也可以被视为两个维度的重力模型。LQG曲面具有许多有趣和令人惊讶的几何性质，并连接到各种其他数学对象，包括各种类型的随机曲线、随机排列和随机图。这个项目将解决该主题中一些最重要的公开问题。获奖者还将参与外展、指导和传播活动，包括指导参与研究的本科生和研究生，撰写说明性文章，组织会议，并在会议、研讨会和暑期学校提供演讲和小型课程。该项目的目标之一是更好地了解LQG曲面中点之间的距离。另一个目标是理解处于所谓的超临界阶段(对应于中心电荷在1和25之间)的LQG表面，这一阶段比亚临界阶段了解得少得多，但从弦理论的角度来看，预计会更有趣。获奖者还计划研究LQG表面上偏微分方程解的意义程度，并进一步探索LQG在组合对象中的应用，包括平面地图、排列和曲折。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。

This project will study Liouville quantum gravity (LQG), a theory of random surfaces. These random surfaces arise in various areas of theoretical physics. In bosonic string theory, they can be used to model the surfaces swept out by a string moving through space. They can also be viewed as models of gravity in two dimensions. LQG surfaces have many interesting and surprising geometric properties and are connected to a variety of other mathematical objects, including various types of random curves, random permutations, and random graphs. This project will address some of the most important open problems in the subject. The awardee will also engage in outreach, mentorship, and dissemination activities, including mentoring undergraduate and graduate participation in the research, writing expository articles, organizing conferences, and giving talks and mini-courses at conferences, seminars, and summer schools. One goal of the project is to build a better understanding of distances between points in LQG surfaces. Another goal is to understand LQG surfaces in the so-called supercritical phase (corresponding to central charge between 1 and 25), which is much less well understood than the subcritical phase, but which is expected to be more interesting from a string theory perspective. The awardee also plans to study the extent to which one can make sense of the solutions to partial differential equations on LQG surfaces, and to further explore the applications of LQG to combinatorial objects, including planar maps, permutations, and meanders.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.