Probabilistic Combinatorics

该项目侧重于一些相互关联的问题，所有这些问题本质上都是组合的（特别是它们通常处理有限结构），但与其他领域有联系。尽管PI的许多工作都与其他学科有关——其中一些已经产生了意想不到的应用后果——但这里的重点是从数学的角度来看最有趣的东西。长期以来，PI一直对跨越数学界限的工作感兴趣。他成功地应用了其他数学学科（代数、几何、拓扑学、概率论、傅立叶分析、信息论）的思想来解决众所周知的组合问题，并将组合思想应用于其他领域（如几何、计算机科学、概率论、统计力学）的问题。就像在这个项目中一样，他最感兴趣的是那些简单但看似基本的问题，这些问题有着抗拒解决的历史，尤其是他认为解决这些问题必然会迫使一个人超越现有的方法。此外，该项目还为研究生提供了研究培训机会。该项目处理组合主题，这些主题代表了PI当前的一些主要兴趣，其中大多数具有一些概率方面。一种常见的设置是有限集合的子集的集合，或者（以不同的方式看待同一对象）给定长度的二进制字符串的“汉明立方体”。概率性更强的问题主要与阈值和/或事件之间的依赖性有关；概率较小的则属于“极值”组合学。所有这些都涉及几十年来在各自领域一直处于核心地位的概念。例如，阈值——粗略地说，是随机系统中各种现象首次出现的强度——自1960年以来一直是概率组合学和统计物理相关部分的核心，也是PI近年来最一致的关注点。集合系统和等周问题的相交性质（这些是概率较小的主题）具有相似的历史和中心性。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。

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.