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

许多在局部粗糙化的情况下表现出全局平滑的增长模型被预测为与典型的非线性随机偏微分方程（称为kardar - paris - zhang （KPZ）方程）相同的方式。尽管一些引人注目的代数对象的双射，如随机矩阵、杨氏图等，在一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还不足以分析这些系统的一些基本几何性质。PI继续将可积方法与主要的概率和几何角度结合起来，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开创新的研究方向。该项目还具有很强的教育成分，包括指导研究生和博士后，以及本科和研究生阶段的课程开发，旨在创建多个高级研究主题课程，并设计本科基础课程中的概率内容。各种教育和传播策略，包括举办讲习班，撰写调查文章，教授暑期课程。在KPZ通用性类中预测的一个典型的生长模型是平面最后通道渗透（LPP）模型，它在平面晶格的顶点上随机赋值，并考虑顶点之间的路径，其中权重最大。这种称为测地线的最大路径是研究的基本对象。除了进一步发展LPP模型中测地线合并的图像外，该项目旨在研究路径的整个能量景观及其相关权重，重点关注几乎最大路径或近测地线的几何形状。PI还将研究测地线西瓜的交错特性（具有最大累积重量的不相交路径的集合）及其与各种线集合和决定性点过程的连接的后果，以及测地线的大偏差行为。最后，该项目将启动有关分形几何和噪声敏感性的新研究方向，从临界平面渗透和自旋玻璃的背景下获得类似性质的开创性作品的灵感。特定的主题包括研究各种端点对的豪斯多夫维，这些端点对承认特殊的测地线行为，以及计算标志着LPP自然动态版本中混沌开始的指数。作为必要的工具，几个新的理论，包括离散谐波分析模型在KPZ普适类将发展。特别是，它有望在概率、数学物理和理论计算机科学的各个社区之间建立一座桥梁。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。

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.