CAREER: Set-Systems: Probabilistic, Geometric and Extremal Perspectives

在这个项目中，PI将研究离散数学和概率的各种主题，这两个领域在21世纪的深度和广度都有了显著的增长，产生了适用于许多科学学科的方法。这些包括理论计算机科学和统计物理的应用和许多重大突破，更不用说数学的其他一些领域了。该项目产生的方法和技术将对这些领域的发展产生重大影响，也将适用于数学和理论计算机科学的其他分支。该项目还提出了培养本科生和研究生的综合计划。该项目的广泛研究目标包括从组合、概率和几何的不同角度研究布尔超立方体中的集合系统。上集和下集特别是在许多不同的数学领域中作为基本的构件出现：它们编码概率空间中的许多感兴趣的事件，它们产生于概率组合学和统计物理学，它们出现在极值集合论中，相交族提供了一类重要的例子，它们作为半空间的离散类似物出现在凸几何中，并被视为单纯形复形，它们是单纯拓扑学中的中心对象。鉴于这些物体在多个领域的普遍存在，更好地了解它们的各个方面将推动这些领域的发展。通过仔细选择来自组合学、概率和几何的关于集合系统的基本问题，PI旨在开发新的分析、组合、概率和几何技术来研究这些对象。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。

In this project, the PI will study various topics in discrete mathematics and probability, two areas which have grown significantly in both depth and breadth in the 21st century, resulting in methods that apply to a number of scientific disciplines. These include applications, and many significant breakthroughs, in theoretical computer science and statistical physics, not to mention a number of other areas of mathematics. The approaches and techniques resulting from this project will have a significant impact on the development of these areas and will also be applicable in other branches of mathematics and theoretical computer science. This project also puts forward a comprehensive plan for training undergraduate and graduate students.The broad research goals of this project involve studying set-systems in the Boolean hypercube from various points of view: combinatorial, probabilistic and geometric. Up-sets and down-sets in particular appear as basic building blocks in a number of different areas of mathematics: they encode many events of interest in probability spaces arising in probabilistic combinatorics and statistical physics, they arise in extremal set theory, with intersecting families furnishing an important class of examples, they appear in convex geometry as discrete analogues of half-spaces, and viewed as simplicial complexes, they are central objects in simplicial topology. Given the prevalence of these objects across multiple areas, a better understanding of their various facets will spur developments across these fields. By targeting a careful selection of basic problems about set-systems arising from combinatorics, probability and geometry, the PI aims to develop new analytic, combinatorial, probabilistic and geometric techniques to study these objects.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.