Bijective Combinatorics for Geometrical Structures

这个项目是在组合学领域，它是数学的一个分支，涉及离散数据结构的描述和分析。组合分析是解决数学中许多问题的中心工具，也是计算机科学、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了一些无法解释的简单模式。这表明，这些结构可能有另一种更简单的描述，即“双射”，来解释这些模式。找到这样的双射可以极大地简化对所考虑对象的数学分析。研究生将作为这个项目的一部分接受培训。这个项目的目标是使用双射技术解决数学中的几个重要问题。第一个问题是关于平面映射和晶格路径之间的一类双射的统一和推广，目的是分析随机晶格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。第二个问题受图论的启发，旨在寻找正确着色平面三角剖分的神秘计数恒等式背后的组合结构。第三个项目的目的是将Schnyder Wood推广到平面图上，以设计一些图形绘制算法。其他项目涉及超图的组合学，Coxeter超平面排列中的人脸编码，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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.