Combinatorics of discrete surfaces

Brandeis University, Waltham MA

Investigators

Abstract

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.