Combinatorial geometry in discrete math and harmonic analysis

离散数学和调和分析中的组合几何

• 批准号: RGPIN-2017-03865
• 负责人: Zahl, Joshua
• 金额: $ 3.64万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758606
• 关键词: Combinatorial; geometry; discrete; math; harmonic

Over the past 25 years, harmonic analysis and combinatorics have become increasingly entwined. Jean Bourgain, Nets Katz, Wilhelm Schlag, Terence Tao, Tom Wolff, and many others have discretized harmonic analysis problems and recast them as questions in combinatorial geometry. In the opposite direction, the recent and growing field of discrete analysis uses tools and ideas from analysis to attack problems that have traditionally been the purview of combinatorics. Broadly speaking, my work lies at the intersection of geometric combinatorics and classical harmonic analysis, and sprawls in several related directions.My research program is guided by two major open problems: the Kakeya problem and the Erdos distinct distance problem. The first of these problems concerns objects called Besicovich sets. A Besicovich set in d dimensions is a subset of d-dimensional space that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that sets of this type must be "large" in a certain technical sense. Today, the Kakeya problem plays a central role in the interactions between classical harmonic analysis, theoretical computer science, and classical algebraic geometry. The Erdos distinct distance problem asks: given a set of n points in Rd, how many different distances can occur between pairs of points? This problem succinctly encapsulates a vast region of combinatorial geometry that is poorly understood. In 2010, Larry Guth and Nets Katz obtained nearly tight bounds for the d=2 version of the problem and in doing so, they developed a new set of tools that are leading to sweeping changes across combinatorial geometry, harmonic analysis, and theoretical computer science. In higher dimensions the problem remains open, and settling it has become a central priority for the combinatorial geometry community.Incidence geometry is a field of extremal combinatorics that analyzes how many intersections (called incidences) can occur within a collections of objects. This has proved to be a powerful and flexible framework for describing many types of phenomena, including the two problems discussed above. A central theme running through my research is the role played by algebraic structure in incidence geometry. If a collection of objects is arranged at random, then there are likely to be few, if any, incidences. This suggests that the most problematic (and thus interesting) arrangements are highly structured. So far, we know of two major sources of interesting structures. The first comes from algebraic geometry, and the second comes from sets that are almost closed under arithmetic operations such as addition and multiplication. If both of these sources of interesting structure are absent, then it seems reasonable to expect a decrease in the number of incidences, which would yield stronger results in the original problem. My work focuses on turning this principle into quantitative results.

在过去的25年里，谐波分析和组合学变得越来越复杂。Jean Bourgain，Nets Katz，Wilhelm Schlag，Terence Tao，Tom Wolff和许多其他人已经将调和分析问题离散化，并将其重新转换为组合几何中的问题。在相反的方向，最近和不断增长的离散分析领域使用分析工具和思想来解决传统上属于组合学范围的问题。广义地说，我的工作是几何组合学和经典调和分析的交叉点，并在几个相关的方向蔓延。我的研究计划是由两个主要的开放问题：Kakeya问题和鄂尔多斯不同的距离问题。这些问题中的第一个涉及称为Besicovich集的对象。d维贝西科维奇集是d维空间的一个子集，它包含指向每个方向的单位线段。Kakeya猜想断言这种类型的集合在某种技术意义上必须是“大”的。今天，挂谷问题在经典调和分析、理论计算机科学和经典代数几何之间的相互作用中起着核心作用。鄂尔多斯不同距离问题的问题是：给定Rd中的一组n个点，点对之间可以出现多少不同的距离？这个问题简洁地概括了一个广大的组合几何领域，这是知之甚少。2010年，Larry Guth和Nets Katz获得了d=2版本问题的几乎紧边界，并在此过程中开发了一套新的工具，导致组合几何，调和分析和理论计算机科学的全面变化。在更高的维度上，这个问题仍然是开放的，解决它已经成为组合几何社区的中心优先事项。关联几何是极值组合数学的一个领域，它分析在一个对象集合中可以发生多少相交（称为关联）。事实证明，这是一个强大而灵活的框架，可以用来描述许多类型的现象，包括上面讨论的两个问题。贯穿我研究的一个中心主题是代数结构在关联几何中所起的作用。如果一个物体的集合是随机排列的，那么很可能很少有（如果有的话）发生。这表明，最有问题的（因此也是最有趣的）安排是高度结构化的。到目前为止，我们知道有趣结构的两个主要来源。第一个来自代数几何，第二个来自在算术运算（如加法和乘法）下几乎闭合的集合。如果这两个有趣结构的来源都不存在，那么似乎可以合理地预期事件的数量会减少，这将在原始问题中产生更强的结果。我的工作重点是将这一原则转化为量化结果。