Liouville Quantum Gravity and Its Applications

这个项目将研究刘维尔量子引力（LQG），一种随机表面理论。这些随机表面出现在理论物理的各个领域。在玻色子弦理论中，它们可以用来模拟弦在空间中运动时扫出的表面。它们也可以被看作是二维重力模型。LQG曲面具有许多有趣和令人惊讶的几何特性，并与各种其他数学对象相连，包括各种类型的随机曲线、随机排列和随机图。这个项目将解决这个主题中一些最重要的开放性问题。获奖者还将参与外展、指导和传播活动，包括指导本科生和研究生参与研究、撰写说明性文章、组织会议、在会议、研讨会和暑期学校发表演讲和迷你课程。该项目的一个目标是更好地理解LQG表面上点之间的距离。另一个目标是了解所谓的超临界相（对应于1到25之间的中心电荷）的LQG表面，这比亚临界相了解得少得多，但从弦理论的角度来看，它预计会更有趣。获奖者还计划研究人们在多大程度上可以理解LQG曲面上的偏微分方程的解，并进一步探索LQG在组合对象上的应用，包括平面图、排列和曲线图。该奖项反映了美国国家科学基金会的法定使命，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 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.