AF: Small: Understanding Expansion Phenomena: Graphical, Hypergraphical, Geometric, and Quantum

AF：小：理解膨胀现象：图形、超图形、几何和量子

• 批准号: 2326685
• 负责人: Madhur Tulsiani
• 金额: $ 20.22万
• 依托单位: Toyota Technological Institute at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-01 至 2025-09-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2326685&HistoricalAwards=false
• 关键词: AF; Small; Understanding; Expansion; Phenomena

A broad variety of phenomena in computation can be viewed as different forms of “expansion”, which ensure that local properties which can be observed by looking at small parts of an object, can be used to influence and understand global properties exhibited at a much larger scale. This is an important design requirement in several applications, such as (1) classical and quantum error-correction, where one wants errors to be easily detectable by local checks, (2) optimization problems, where one wants local choices to push the global solution towards optimality, and (3) geometric embeddings of high-dimensional data, where one wants to use local (low-dimensional) conditions to influence high-dimensional behavior. In the past few years, several new concepts and techniques have emerged to study expansion phenomena in different contexts. This project aims to study several different forms of expansion phenomena in a unified way, with an emphasis on applications in the areas of error-correcting codes and (approximate) optimization. This research is likely to lead to new connections between multiple areas where such phenomena are useful. The material generated as part of this research will also be disseminated through surveys and a series of expository videos. This project aims to obtain a unified view of the following different forms and applications of expansion phenomena:- Classical notions of graph expansion and novel notions of high-dimensional expansion for hypergraphs, and their connections to recent advances in coding theory.- Applications of classical expansion phenomena to quantum codes, as well as quantum extensions of classical expansion phenomena.- Connections of high-dimensional expansion to the study and approximability of expansion phenomena in geometric spaces, and related problems about fine-grained graph expansion.The research directions pursued in this project aim to introduce new techniques in algorithmic coding theory and in the study of approximability of discrete and continuous optimization problems. The project considers several problems that have proved to be bottlenecks for current algorithmic and analytic techniques, explores new approaches arising from the study of expansion in a different context. The project aims to apply these ideas for the design of new error-correcting codes, and new algorithms for existing codes, towards the design of new pseudorandom objects, and also new families of combinatorial and geometric instances for proving inapproximability results.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.

计算中的各种现象可以被视为不同形式的“扩展”，这确保通过观察对象的小部分可以观察到的局部属性可以用来影响和理解在更大范围内表现出的全局属性。这是多种应用中的一项重要设计要求，例如（1）经典和量子纠错，人们希望通过局部检查轻松检测到错误；（2）优化问题，人们希望局部选择将全局解决方案推向最优；（3）高维数据的几何嵌入，人们希望使用局部（低维）条件来影响高维行为。在过去的几年中，出现了一些新的概念和技术来研究不同背景下的膨胀现象。该项目旨在以统一的方式研究几种不同形式的扩展现象，重点是纠错码和（近似）优化领域的应用。这项研究可能会在这些现象有用的多个领域之间建立新的联系。作为本研究的一部分生成的材料也将通过调查和一系列说明视频进行传播。该项目旨在获得对膨胀现象的以下不同形式和应用的统一视图：-图膨胀的经典概念和超图高维膨胀的新概念，以及它们与编码理论最新进展的联系。-经典膨胀现象在量子代码中的应用，以及经典膨胀现象的量子扩展。-高维膨胀与研究和近似性的联系 几何空间中的扩展现象，以及细粒度图扩展的相关问题。该项目的研究方向旨在引入算法编码理论以及离散和连续优化问题的逼近性研究中的新技术。该项目考虑了几个已被证明是当前算法和分析技术瓶颈的问题，探索了不同背景下扩展研究中产生的新方法。该项目旨在将这些思想应用于新的纠错码的设计、现有代码的新算法、新的伪随机对象的设计以及新的组合和几何实例系列，以证明不可近似性结果。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。