Bijective Combinatorics for Geometrical Structures

几何结构的双射组合

• 批准号: 2154242
• 负责人: Olivier Bernardi
• 金额: $ 21万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154242&HistoricalAwards=false
• 关键词: Bijective; Combinatorics; Geometrical; Structures

This project is in the field of combinatorics, which is a branch of mathematics concerned with the description and analysis of discrete data structures. Combinatorial analysis is a central tool to solve many problems in mathematics, and in related fields such as computer science and mathematical physics. This project includes several problems in which some seemingly complex structures seem to display some unexplained simple patterns. This suggest that these structures may have an alternative simpler description, a "bijection", explaining these patterns. Finding such a bijection can greatly simplify the mathematical analysis of the objects under consideration. Graduate students will be trained as part of this project.This projects aims at solving several important problems in mathematics using bijective techniques. The first problem is about unifying and extending a class of bijections between planar maps and lattice paths, in view of analyzing some statistical mechanics models on random lattices and relating them to Liouville quantum gravity and SLE curves. The second problem, motivated by graph theory, aims at finding the combinatorial structures underlying a mysterious enumerative identity for properly colored planar triangulations. The third project aims at using a generalization of Schnyder woods for planar graphs in order to design some graph-drawing algorithms. Other projects are about the combinatorics of hypergraphs, the encoding of faces in Coxeter hyperplane arrangements, and the enumeration of simplicial trees.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.

这个项目是在组合学领域，这是数学的一个分支，与离散数据结构的描述和分析有关。组合分析是解决数学中许多问题的核心工具，也是计算机科学和数学物理等相关领域的核心工具。这个项目包括几个问题，其中一些看似复杂的结构似乎显示出一些无法解释的简单模式。这表明，这些结构可能有一个替代的更简单的描述，一个“双射”，解释这些模式。找到这样一个双射可以大大简化所考虑对象的数学分析。研究生将作为该项目的一部分进行培训。该项目旨在使用双射技术解决数学中的几个重要问题。第一个问题是统一和推广一类平面映射与格点路径之间的双射，这是为了分析随机格点上的一些统计力学模型，并将它们与Liouville量子引力和SLE曲线联系起来。第二个问题，图论的动机，目的是找到一个神秘的枚举身份正确着色的平面三角形的组合结构。第三个项目的目的是使用平面图的推广，以设计一些图形绘制算法的施奈德伍兹。其他项目是关于超图的组合学，Coxeter超平面安排中的面编码，以及单纯树的枚举。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。