Combinatorics of discrete surfaces

本研究项目主要研究离散曲面，即多边形沿边缘粘合而成的曲面。离散曲面（也称为映射）是基本的数学结构，在数学和理论物理的几个分支中起着重要作用，在计算机科学（计算机图形学和电路布局）中也具有重要的实际意义。该项目的目标是通过简单的数学结构（如树和点阵路径）找到地图的编码来解决关于离散曲面的几个开放问题。特别是，这些编码应使用来研究大型随机离散曲面的性质，并设计离散曲面的算法。该项目的一些更广泛的影响是指导本科生的研究项目，组织每周的研讨会和会议，并通过调查文章传播科学发现。PI打算用一种客观的方法解决与地图有关的几个问题。近年来，在地图类和简单的数学结构（如树）之间发现了许多对偶。这代表了对地图理解的突破，它帮助解决了以前无法解决的问题，例如描述随机地图和随机曲面的度量属性。在与Eric Fusy的合作中，PI发现了一个“主双射”，它概括并统一了大多数已知的地图双射。PI计划在以下方向继续他的研究：-将主双射方法扩展到新的地图类别，即超地图和不可约地图。这将完成地图双射理论的统一，并将赋予地图统计力学模型的双射包含在内。-利用最近关于“带电映射”的双射结果研究随机映射的度量性质。写一篇关于地图的客观方法的调查文章。-设计绘图算法的图形使用规范的方向。PI的其他与映射没有直接关系的目标是对Schur测度的Okounkov行列式公式的组合证明，以及支化聚合物空间的参数化。

This research project focuses on the study of discrete surfaces, that is, surfaces obtained by gluing together polygons along edges. Discrete surfaces (also known as maps) are fundamental mathematical structures which play an important role in several branches of mathematics, and theoretical physics and are also of practical importance in computer science (for computer graphics and circuit layout). The goal of the project is to solve several open problems about discrete surfaces by finding encodings of maps by simpler mathematical structures such as trees and lattice paths. In particular, these encodings shall be used to study the properties of large random discrete surfaces, and to design algorithms for discrete surfaces. Some broader impacts of the project are the mentoring of undergraduate research projects, the organization of weekly seminars and conferences, and the dissemination of scientific discoveries through survey articles.The PI intends to solve several problems related to maps using a bijective approach. In recent years many bijections were found between classes of maps and simpler mathematical structures such as trees. This represented a breakthrough in the understanding of maps, which helped solve previously unreachable problems such as describing the metric properties of random maps and random surfaces. In a collaboration with Eric Fusy, the PI has found a "master bijection" generalizing and unifying most known bijections for maps. The PI plans to pursue his research in the following directions:- Extend the master bijection approach to new classes of maps, namely, hypermaps and irreducible maps. This will complete the unification of the bijective theory of maps, and encompass bijections for maps endowed with a statistical mechanics model.- Study the metric properties of random maps by using a recent bijective result about ``charged maps''. - Write a survey article about the bijective approach to maps. - Design drawing algorithms for graphs using canonical orientations. Other goals of the PI which are not directly related to maps are a combinatorial proof of Okounkov determinantal formula for Schur measures, and the parametrization of the space of branched polymers.