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Bijective Approach to Discrete Geometries

离散几何的双射方法

基本信息

批准号：
1800681
负责人：
Olivier Bernardi
金额：
$ 15万
依托单位：
Brandeis University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800681&HistoricalAwards=false
关键词：
Bijective Approach Discrete Geometries

项目摘要

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.

组合数学是数学的一个中心分支，主要研究离散数据结构的描述和分析。组合主义者试图揭示这种结构中的模式和积木，以解释它们的全局行为。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核心重要性。这个项目的重点是编码的几个离散的几何结构（平面图，多面体分解的空间等）。更简单的数学结构。这样的编码，它提供了一个真正不同的描述相同的结构，可以大大简化分析对象的考虑。实际上，隐藏在原始描述中的某些模式和概率行为可能在替代描述中更清楚地出现。该项目旨在开发双射工具，以解决组合学中的几个基本开放问题，其动机来自概率，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双射赋予的平面三角剖分的一些二维格行走。这个项目的其他目标与图的适当着色有关（用双射解释恰当着色平面图的一个计数公式），对称群的可积系统方法（“对角化”KP微分方程支配对称群中的因式分解），超平面安排（Coxeter布置的变形的面的双射），和图形绘制算法（Schnyder woods和transverse structures的同时推广）该奖项反映了NSF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进行评估，被认为值得支持审查标准。