Combinatorics of discrete surfaces

本研究项目主要研究离散曲面，即通过将多边形沿边缘粘合而得到的曲面。离散曲面(也称为地图)是基本的数学结构，它在数学和理论物理的几个分支中扮演着重要的角色，在计算机科学(计算机图形学和电路布局)中也具有实际意义。该项目的目标是通过寻找更简单的数学结构(如树和格子路径)的映射编码来解决几个关于离散曲面的公开问题。特别是，这些编码应用于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，每周组织研讨会和会议，以及通过调查文章传播科学发现。PI打算用双射方法解决与地图有关的几个问题。近年来，在映射类和更简单的数学结构(如树)之间发现了许多双射。这代表了对地图理解的突破，它帮助解决了以前无法达到的问题，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“大师双射”，它概括和统一了地图上最已知的双射。PI计划在以下方面继续他的研究：-将主双射方法扩展到新的映射类，即超映射和不可约映射。这将完成映射的双射理论的统一，并包含具有统计力学模型的映射的双射。-利用最近关于“带电映射”的双射结果来研究随机映射的度量性质。-写一篇关于地图的双射方法的调查文章。-为使用规范方向的图形设计绘图算法。PI的其他与映射没有直接关系的目标是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.