Final Report Summary - EXPLOREMAPS (Combinatorial methods, from enumerative topology to random discrete structures and compact data representations.) The ExploreMaps grant has allowed to set up within the computer science laboratory of Ecole Polytechnique a leading research group in combinatorics and algorithms. The focus of the group is on the notion of combinatorial maps (aka combinatorial descriptions of 2d surfaces), and our work embrace problems arising in a wide range of contexts: statistical physics, data compression, enumerative topology, etc. Our methodology resorts mainly to algorithmics and enumerative combinatorics, but also to probability and spectral theory. The main achievements of the project has been the development of a bijective theory of enumeration of maps on surfaces and its applications. The principle stated in the original proposal, that “classical exploration algorithms applied to maps as opposed to graphs lead to remarkable and unexpected "context-free" decompositions of discrete surfaces”, has been confirmed in many ways: from the original collection of examples that motivated the proposal, a unified framework has emerged, the setting has been extended from the plane to higher genus and to non-orientable surfaces. Finally a parallel theory was developed for combinatorial models of Riemann surfaces, leading to new insights on the famous Hurwitz simple and double numbers.We have used some of these combinatorial results to design new graph drawing and random sampling algorithms, and to propose new encodings and data structures for the succinct representation of 2d geometric objects. Some achievements of the project also regard recent probabilistic resultson random discrete surfaces, in particular in the study of the continuum limit of large random maps.
