CartesEtPlus

CartesEtPlus (French for MapsAndMore) is a 5-year combinatorics project. It is centered on enumerative combinatorics and bijective methods, but has strong interactions with algebraic combinatorics, probability, geometry, and theoretical physics. The main objects of study are discrete two-dimensional topological objects, and in particular maps, which describe the embedding of a graph into a surface, and related objects such as meanders or multicurves. Even if the theme of maps, and in particular planar maps, is well established, new directions are emerging at the boundaries of the subject with physics and geometry. These have not yet been studied in detail according to the combinatorial approach, even though they bring new insights and raise profound challenges. The project will seek to unify these emerging themes within combinatorics of maps, in order to bring out new directions, to share new tools between the different fields addressed, and solve important problems by combining these viewpoints and approaches. The project overarches the three following themes: A/ The structural decomposition of maps; B/ Enumerative problems linked to meanders and multicurves; C/ New models of geometry, from maps of large genus to multicurves or nonorientable surfaces. The project is structured into four precise sets of tasks: 1/ Structural decompositions (bijective constructions for the topological recursion or KP hierarchy, Voronoy tesselations) 2/ Meanders and Multicurves (combinatorial approaches, study of new meanders variants) 3/ New models of Random Geometry (random high-genus maps, random multicurves) 4/ Generating functions (catalytic variables, symmetric functions). The consortium is made from 18 active researchers, mostly from the core French combinatorics community, but it also includes leading experts from theoretical physics and geometry at the origin of the emerging interfaces.