Polynomial Methods in Discrete Geometry

离散几何中的多项式方法

• 批准号: 1710305
• 负责人: Adam Sheffer
• 金额: $ 15.54万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2017-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1710305&HistoricalAwards=false
• 关键词: Polynomial; Methods; Discrete; Geometry

Starting around 2009, a series of difficult longstanding combinatorial problems have been solved by using algebraic techniques. This led to a new subfield that is sometimes called "the Polynomial Method". The philosophy behind this subfield is that collections of objects that exhibit an extremal behavior often have hidden algebraic structure. As a simple example, if a finite set of points in the plane contains many pairs that are at a distance of 1 from each other, one might expect this set to have a grid structure. Once the algebraic structure is found, it can be exploited to gain a better understanding of the original problem. To expose the algebraic structure, one defines polynomials on the studied objects and explores their properties by using algebraic tools, often from Algebraic Geometry. For example, for a problem involving a finite set of points in the plane, one might wish to study the properties of a minimum-degree polynomial that vanishes on the point set. Beyond solving various longstanding problems, this new subfield is also leading to the discovery of interesting connections between Combinatorial Geometry and other fields, such as Harmonic Analysis and Theoretical Computer Science. This project is based on the assumption that the new "algebraic era" in Combinatorial Geometry has not yet reached its peak. It investigates further problems that seem approachable via algebraic techniques, and aims to further develop the current algebraic tools.The first part of this project involves studying several main open problems in Discrete Geometry by using algebraic methods. Specifically, studying the distinct distances problem in R^d, the characterization of point sets that determine few distinct distances, and the problem of deriving stronger and more general bounds for incidence problems. Part of the importance of deriving stronger incidence bounds is that many problems can be reduced to incidence problems (including problems from Combinatorics, Harmonic Analysis, and Number Theory). The second part of the project concerns connections between Discrete Geometry and Additive Combinatorics, also by using polynomial methods. This partly involves further studying known connections between the two fields, but focuses mainly on establishing a new type of connection. This connection consists of defining geometric variants of additive energy and extending some of the known additive energy results to these variants (such as the Balog-Szemeredi-Gowers theorem and higher moment energies).

从2009年开始，一系列长期存在的困难组合问题已经通过使用代数技术得到解决。这导致了一个新的子领域，有时被称为“多项式方法”。这个子领域背后的哲学是，表现出极端行为的对象集合通常具有隐藏的代数结构。举一个简单的例子，如果平面上的一个有限点集包含许多彼此距离为1的点对，人们可能会认为这个点集具有网格结构。一旦找到了代数结构，就可以利用它来更好地理解原始问题。为了揭示代数结构，人们在所研究的对象上定义多项式，并通过使用代数工具（通常来自代数几何）来探索它们的属性。例如，对于涉及平面上有限点集的问题，人们可能希望研究在点集上为零的最小次数多项式的性质。除了解决各种长期存在的问题，这个新的子领域也导致发现组合几何和其他领域之间的有趣联系，如调和分析和理论计算机科学。这个项目是基于这样的假设，即新的“代数时代”在组合几何尚未达到顶峰。它探讨了进一步的问题，似乎可以通过代数技术，并旨在进一步发展目前的代数工具。本项目的第一部分涉及研究几个主要的开放问题，离散几何使用代数方法。具体来说，研究R^d中的不同距离问题，确定几个不同距离的点集的特征，以及推导关联问题的更强和更一般的边界的问题。推导出更强的关联边界的部分重要性在于，许多问题可以简化为关联问题（包括组合数学、调和分析和数论中的问题）。该项目的第二部分涉及离散几何和加法组合学之间的联系，也使用多项式方法。这部分涉及进一步研究这两个领域之间的已知联系，但主要集中在建立一种新的联系。这种联系包括定义加性能量的几何变量，并将一些已知的加性能量结果扩展到这些变量（如Balog-Szemeredi-Gowers定理和更高的矩能量）。