Bijective Approach to Discrete Geometries

组合学是数学的一个中心分支，涉及对离散数据结构的描述和分析。组合学家试图揭示这种结构中的模式和组成部分，以解释它们的整体行为。因此，组合工具在许多其他科学领域（如计算机科学、统计力学、统计学和概率论）中具有中心重要性。这个项目的重点是用更简单的数学结构编码几个离散的几何结构（平面图，空间的多边形分解等）。这种编码为相同的结构提供了真正不同的描述，可以极大地简化对所考虑的对象的分析。事实上，隐藏在原始描述中的某些模式和概率行为，可能会在替代描述中更清晰地出现。该项目旨在开发客观的工具，以解决组合学中的几个基本开放问题，其动机来自概率论，理论物理和计算机科学。其中一个主要目标是为三个非常重要的概率结构之间的深层关系的多作者证明奠定基础：随机平面图，高斯自由场和SLE曲线。证明将建立在一个双射编码的渗透赋予平面三角形的一些二维点阵行走。这个项目的其他目标与图的适当着色（解释适当着色的平面图的双射计数公式），对称群的可积系统方法（对角化控制对称群中分解的KP微分方程），超平面排列（Coxeter排列变形面的双射），和图形绘制算法（施耐德森林和横向结构的同时泛化）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Combinatorics is a central branch of mathematics concerned with the description and analysis of discrete data structures. Combinatorialists try to uncover patterns and building blocks in such structures, in order to explain their global behavior. Combinatorial tools are therefore of central importance in many other fields of science, such as computer science, statistical mechanics, statistics, and probability. This project focuses on the encoding of several discrete geometrical structures (planar graphs, polytopal decompositions of space, etc.) by simpler mathematical structures. Such an encoding, which provides a genuinely different description of the same structures, can greatly simplify the analysis of the objects under consideration. Indeed, certain patterns and probabilistic behavior which were hidden in the original description, may appear more clearly in the alternative description.This projects aims to develop bijective tools in order to solve several fundamental open problems in combinatorics, with motivations coming from probability, theoretical physics, and computer science. One of the major goals is to set the foundation for a multi-authored proof of a deep relation between three very important probabilistic constructions: random planar graphs, the Gaussian free field, and SLE curves. The proof will be built upon a bijective encoding of percolation-endowed planar triangulations by some two-dimensional lattice walks. Other goals of this project are related to proper coloring of graphs (explaining bijectively a counting formula for properly colored planar graphs), integrable system approach to the symmetric group ("diagonalizing" the KP differential equations governing the factorizations in the symmetric group), hyperplane arrangements (bijections for the faces of the deformations of the Coxeter arrangement), and graph drawing algorithms (simultaneous generalization of Schnyder woods and transversal structures).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.