Polynomial Methods in Discrete Geometry

从2009年左右开始，使用代数技术解决了一系列长期存在的组合问题。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。举个简单的例子，如果平面上的一个有限点集包含许多对，它们彼此之间的距离为1，那么人们可能会期望这个集合具有网格结构。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，人们在研究对象上定义多项式，并使用代数工具（通常来自代数几何）探索它们的性质。例如，对于一个涉及平面上有限点集的问题，人们可能希望研究在点集上消失的最小次多项式的性质。除了解决各种长期存在的问题之外，这个新的子领域还导致发现组合几何与其他领域之间的有趣联系，例如谐波分析和理论计算机科学。这个项目是基于一个假设，即组合几何的新“代数时代”尚未达到顶峰。它进一步研究了似乎可以通过代数技术接近的问题，并旨在进一步发展当前的代数工具。本课题的第一部分涉及到用代数方法研究离散几何中的几个主要开放问题。具体来说，研究了R^d中的不同距离问题，确定几个不同距离的点集的表征，以及为关联问题导出更强和更一般的界的问题。推导更强的关联界的部分重要性在于，许多问题可以简化为关联问题（包括来自组合学、调和分析和数论的问题）。项目的第二部分涉及离散几何和加性组合学之间的联系，也是通过使用多项式方法。这部分涉及进一步研究这两个领域之间已知的联系，但主要侧重于建立一种新的联系。这种联系包括定义可加能量的几何变体，并将一些已知的可加能量结果扩展到这些变体（如balog - szemeredii - gowers定理和更高的矩能）。

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).