Random Surfaces and Related Questions

本研究涉及随机曲面、随机树和随机曲线的研究。这些对象与许多不同的数学领域（如组合数学，复分析，图论和随机矩阵理论）和许多不同的物理领域（如弦理论，规范理论，粒子物理和二维统计力学）有关。 PI将帮助指导和支持本科生和研究生，以及博士后研究人员。正在研究的具体对象包括Schramm-Loewner演化，连续随机树和几个随机表面模型，包括刘维量子引力，布朗映射，peanosphere和刘维共形场论。 这些随机表面模型在某种意义上都是等价的，但证明和理解它们的等价性一直是数学和物理学中的一项重大任务，并且仍然存在许多悬而未决的问题。具体目标包括更好地理解增长模型和量子Loewner演化，更好地理解表面和规范理论之间的关系，更好地理解高亏格表面，以及更详细地理解相关的晶格模型和随机平面映射。总的来说，该奖项的目标是为物理现象的一些最基本的模型提供更强的数学理解。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。

This research concerns the study of random surfaces, random trees and random curves. These objects are relevant to many different fields of mathematics (such as combinatorics, complex analysis, graph theory, and random matrix theory) and to many different areas of physics (such as string theory, gauge theory, particle physics and two-dimensional statistical mechanics). The PI will help mentor and support both undergraduate and graduate students, as well as postdoctoral researchers.The specific objects under study include Schramm-Loewner evolution, continuum random trees, and several random surface models, including Liouville quantum gravity, the Brownian map, the peanosphere, and Liouville conformal field theory. These random surface models are all in some sense equivalent, but proving and understanding their equivalence has been a major undertaking within both mathematics and physics, and many open problems remain. The specific goals include a better understanding of growth models and quantum Loewner evolution, a better understanding of the relationship between surface sums and gauge theory, a better understanding of high genus surfaces, and a more detailed understanding of the associated lattice models and random planar maps. Overall, the goal is to provide a stronger mathematical understanding of some of the most fundamental models for physical phenomena.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.