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

职业：Kardar-Parisi-Zhang 普遍性的各个几何方面：分形维数、噪声敏感性、线系综和大偏差。

基本信息

批准号：
1945172
负责人：
Shirshendu Ganguly
金额：
$ 40万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945172&HistoricalAwards=false
关键词：
Career Various Geometric Aspects Kardar

项目摘要

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.

许多在存在局部粗糙化的情况下表现出全局平滑的生长模型预计其行为方式与称为 Kardar-Parisi-Zhang (KPZ) 方程的规范非线性随机偏微分方程相同。尽管一些对代数对象的显着双射（例如随机矩阵、杨氏图等）导致了一些预测的数学验证方面的一系列突破，但当前基于可积概率的许多技术本身不足以分析此类系统的一些基本几何性质。继续进行一项将可积方法与主要概率和几何视角相结合的计划，PI 制定了一项全面的计划来研究此类模型的多个方面，这将从根本上提高我们的理解并启动新的研究方向。该项目还具有强大的教育成分，包括指导研究生和博士后，以及本科生和研究生水平的课程开发，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。还将开展各种教育和传播策略，包括组织研讨会、撰写调查文章、教授暑期课程。预计属于 KPZ 普适性类别的规范增长模型是平面最后通道渗透 (LPP) 模型，它将随机权重放在平面晶格的顶点上，并考虑产生最大权重的顶点之间的路径。这种称为测地线的最大路径是研究的基本对象。除了进一步开发 LPP 模型中测地线合并的图景外，该项目还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（具有最大累积权重的不相交路径的集合）及其与各种线系综和行列式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颖研究方向，从临界平面渗滤和自旋玻璃背景下类似性质的开创性工作中汲取灵感。具体主题包括研究各种端点对的豪斯多夫维数，这些端点对承认异常的测地线行为，以及标记 LPP 自然动态版本中混沌开始的计算指数。作为必要的工具，将开发一些新理论，包括 KPZ 通用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领域的各个社区之间建立一座桥梁。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。