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Probabilistic Combinatorics

概率组合学

基本信息

批准号：
1954035
负责人：
Jeffry Kahn
金额：
$ 36万
依托单位：
Rutgers University New Brunswick
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：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954035&HistoricalAwards=false
关键词：
Probabilistic Combinatorics

项目摘要

The project focuses on a number of interrelated problems, all essentially combinatorial (meaning, in particular, that they usually deal with finite structures), but with connections to other areas. Though much of the PI's work has ties to other disciplines---and some of it has had unanticipated applied consequences---the emphasis here is on what seems most interesting from a mathematical standpoint. The PI has long been interested in work that cuts across mathematical boundaries. He has had success both in applying ideas from other mathematical disciplines (algebra, geometry, topology, probability, Fourier analysis, information theory) to settle well-known combinatorial problems, and, on the other hand, in bringing combinatorial ideas to bear on problems from other areas (e.g. geometry, computer science, probability, statistical mechanics). He has---as in this project---most often been interested in simple but seemingly basic questions with histories of resisting solution, motivated especially by the idea that tackling such questions necessarily forces one to go beyond existing methods. In addition the project provides research training opportunities for graduate students.The project treats combinatorial topics representing some of the PI's main current interests, most with some probabilistic aspect. A common setting is the collection of subsets of a finite set, or (the same object viewed differently) the "Hamming cube" of binary strings of a given length. The more probabilistic questions are largely concerned with thresholds and/or dependence among events; the less probabilistic ones belong to "extremal" combinatorics. All involve notions that have been at the hearts of their respective areas for decades. For example, thresholds---roughly, the intensities at which various phenomena first appear in a random system---have been central to probabilistic combinatorics and related parts of statistical physics since about 1960, and the PI's most consistent focus in recent years. Intersection properties of set systems and isoperimetric problems (these are among the less probabilistic topics) are of similar vintage and centrality.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.

该项目专注于一些相互关联的问题，这些问题基本上都是组合的(特别是指它们通常处理有限的结构)，但与其他领域的联系。尽管PI的许多工作与其他学科有联系-其中一些产生了意想不到的应用后果-这里的重点是从数学角度看最有趣的东西。长期以来，PI一直对跨越数学界限的工作感兴趣。他成功地运用了其他数学学科(代数、几何、拓扑、概率、傅立叶分析、信息论)的思想来解决著名的组合问题，另一方面，也成功地将组合思想应用于其他领域(如几何、计算机科学、概率、统计力学)的问题。他-就像在这个项目中-最经常对简单但看似基本的问题感兴趣，这些问题有抵制解决的历史，特别是因为解决这些问题必然会迫使一个人超越现有方法的想法。此外，该项目还为研究生提供了研究培训机会。该项目处理代表PI当前主要兴趣的一些组合主题，大多数带有一些概率方面的内容。一种常见的设置是有限集合的子集的集合，或者(从不同的角度来看相同的对象)给定长度的二进制字符串的“汉明立方体”。概率较高的问题主要与阈值和/或事件之间的相关性有关；概率较低的问题属于“极端”组合学。所有这些都涉及几十年来一直处于各自领域核心的概念。例如，阈值-粗略地说，各种现象在随机系统中首次出现的强度-自1960年以来一直是概率组合学和统计物理学相关部分的核心，也是近年来PI最一致的焦点。集合系统的交集性质和等周问题(这些属于不太可能的主题)具有相似的年代性和中心性。该奖项反映了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。