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年左右开始，通过使用代数技术解决了一系列困难的长期组合问题。这导致了一个新的子场，有时称为“多项式方法”。这个子场的哲学是，表现出极端行为的物体的集合通常具有隐藏的代数结构。作为一个简单的例子，如果平面中的一组有限的点包含许多对彼此距离为1的对，则可能希望该集合具有网格结构。一旦找到代数结构，就可以利用它以更好地了解原始问题。为了揭示代数结构，人们通常使用代数工具（通常来自代数几何形状）来定义所研究对象上的多项式，并通过使用代数工具来探索其属性。例如，对于涉及平面中一组有限点的问题，人们可能希望研究在点集中消失的最低程度多项式的属性。除了解决各种长期存在的问题外，这个新的子领域还导致了组合几何和其他字段之间的有趣联系，例如谐波分析和理论计算机科学。该项目基于以下假设：组合几何形状的新“代数时代”尚未达到顶峰。它研究了通过代数技术看起来很容易平易近人的进一步问题，并旨在进一步开发当前的代数工具。该项目的第一部分涉及通过使用代数方法研究离散几何学中的几个主要开放问题。具体而言，研究r^d的不同距离问题，确定几乎不同距离的点集的表征，以及为发病率问题得出更强，更一般界限的问题。得出更强发生率的重要性的一部分是，许多问题可以简化为发病率问题（包括组合学，谐波分析和数量理论的问题）。该项目的第二部分涉及离散几何形状和添加剂组合物之间的连接，也是使用多项式方法。这部分涉及进一步研究两个字段之间的已知联系，但主要集中于建立新的连接。该连接包括定义添加能的几何变体，并将一些已知的加性能量结果扩展到这些变体（例如Balog-Szemeredi-Gowers定理和更高的力矩能量）。