Probabilistic Combinatorics

该项目的重点是一些相互关联的问题，基本上都是组合的（这意味着，特别是，他们通常处理有限的结构），但与其他领域的连接。 虽然PI的大部分工作与其他学科有联系-其中一些已经产生了意想不到的应用后果-这里的重点是从数学的角度来看似乎最有趣的东西。 PI长期以来一直对跨越数学边界的工作感兴趣。 他已经成功地应用思想从其他数学学科（代数，几何，拓扑，概率，傅立叶分析，信息论）解决著名的组合问题，并在另一方面，在使组合思想承担的问题从其他领域（如几何，计算机科学，概率，统计力学）。 就像在这个项目中一样，他最感兴趣的往往是简单但看似基本的问题，这些问题有着难以解决的历史，特别是出于这样一种想法，即解决这些问题必然会迫使人们超越现有的方法。此外，该项目还为研究生提供了研究培训的机会。该项目涉及代表PI当前主要兴趣的组合主题，其中大多数涉及概率方面。 一个常见的设置是有限集合的子集的集合，或者（从不同的角度看同一个对象）给定长度的二进制字符串的“汉明立方体”。 更多的概率问题主要涉及阈值和/或事件之间的依赖关系;概率较小的问题属于“极值”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，各种现象首次出现在随机系统中的强度-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PI近年来最一致的关注点。 集合系统和等周问题的交叉特性（这些是概率较小的主题）具有类似的历史和中心地位。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.