Career: Various Geometric Aspects of Kardar-Parisi-Zhang Universality: Fractal Dimensions, Noise Sensitivity, Line Ensembles, and Large Deviations.

许多生长模型在存在局部粗糙的情况下表现出全局光滑性，预测其行为与被称为Kardar-Parisi-Zhang(KPZ)方程的正则非线性随机偏微分方程解相同。虽然随机矩阵、杨图等代数对象的双射在某些预测的数学验证方面取得了一系列突破，但目前许多基于可积概率的技术本身并不足以分析这类系统的一些基本几何性质。继续一个正在进行的计划，将可积方法与主要的概率和几何观点相结合，PI制定了一个全面的计划来研究这些模型的几个方面，这将从根本上提高我们的理解并启动新的研究方向。该项目还包括强大的教育组成部分，包括指导研究生和博士后，以及本科生和研究生层面的课程开发，旨在创建多个高级研究主题课程，并设计基础本科课程中的概率内容。还将实施各种教育和传播战略，包括组织讲习班、撰写调查文章、教授暑期课程等。一个被预测为KPZ普适性类的典型增长模型是平面最后通道渗流(LPP)模型，该模型在平面格子的顶点上施加随机权重，并考虑顶点之间获得最大权重的路径。这种称为测地线的最大路径是基本的研究对象。除了进一步发展LPP模型中测地线合并的图景外，该项目还旨在研究路径及其相关权重的整个能量格局，重点是几乎最大路径或接近测地线的几何形状。PI还将研究测地线西瓜(具有最大累积权重的不相交路径的集合)的交错特性及其与各种线系综和行列式点过程的联系的后果，以及测地线的大偏差行为。最后，该计划将启动关于分形几何和噪声敏感性的新研究方向，从临界平面渗流和自旋玻璃背景下类似性质的开创性工作中获得灵感。具体的主题包括研究允许特殊测地行为的各种端点对的Hausdorff维度，以及计算标记LPP自然动力学版本中混沌开始的指数。作为必要的工具，将发展几个新的理论，包括对KPZ普适类模型的离散调和分析。特别是，这有望在概率、数学物理和理论计算机科学的不同社区之间建立一座桥梁。这一奖项反映了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。

Many growth models exhibiting a global smoothing in presence of local roughening are predicted to behave in the same way as a canonical non-linear stochastic partial differential equation known as the Kardar-Parisi-Zhang (KPZ) equation. Although some remarkable bijections to algebraic objects such as random matrices, Young diagrams and so on have led to a series of breakthroughs in the mathematical verification of some of the predictions, many of the current techniques based on integrable probability are not sufficient by themselves to analyze some of the fundamental geometric properties of such systems. Continuing an ongoing program to couple the integrable approach with a primarily probabilistic and geometric perspective, the PI lays down a comprehensive plan to investigate several aspects of such models which will fundamentally improve our understanding and initiate new research directions. The program also has a strong education component including mentoring graduate students and postdocs, and curriculum development at undergraduate and graduate levels, aiming to create multiple advanced research topics courses, and design the probability content in foundational undergraduate courses. Various educational and dissemination strategies including workshop organizing, writing survey articles, teaching summer courses will be carried out as well. A canonical model of growth predicted to be in the KPZ universality class is the model of planar Last Passage Percolation (LPP) which puts random weights on the vertices of a planar lattice and considers paths between vertices which accrue maximum weights. Such maximal paths called geodesics are fundamental objects of study. Besides further developing the picture of coalescence of geodesics in LPP models, the project aims to study the entire energy landscape of paths and their associated weights, with a focus on the geometry of almost maximal paths or near geodesics. The PI will also investigate interlacing properties of geodesic watermelons (collections of disjoint paths with maximal cumulative weight) and their consequences with connections to various line ensembles and determinantal point processes, as well as large deviation behaviors of geodesics. Finally the program will initiate novel research directions concerning fractal geometry and noise sensitivity, drawing inspiration from seminal works of a similar nature in the context of critical planar percolation and spin glasses. Particular topics include studying Hausdorff dimensions of various endpoint pairs admitting exceptional geodesic behavior as well as computing exponents marking the onset of chaos in natural dynamical versions of LPP. As necessary tools, several new theories including discrete harmonic analysis for models in the KPZ universality class will be developed. In particular, this is expected to create a bridge between various communities in probability, mathematical physics and theoretical computer science.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.