Guillaume Chapuy - Pictures.

• Three Random Maps.
  

  They are generated with to the algorithms I developped in my thesis. The 3-dimensional picture is obtained by an algorithm of "spring embedding" of the graph in 3-space. The programs are written in Java and use the Java 3D library. You can use these pictures for academic purposes (but I prefer if you send me an e-mail first). The three maps pictured here have genus 0 (right, top), 1 (top), and 2 (right, bottom).

• Here are pictures I made some time ago to illustrate my research. If you want more up-to-date (but less pictorial) information about my work, look at the preprints and papers on my main page.

• A map on a torus with 2 handles.
  It has 22 edges, 13 vertices, and 7 faces. A combinatorial explanation of the counting exponent 5(g-1)/2 for maps of genus g is given in this paper (with Marcus and Gilles Schaeffer)
  
• A 3-constellation on the torus.
  A constellation is a map that satisfies some additionnal coloring properties. They are in bijection with certain factorisations in the symmetric group. In that paper, I have performed the asymptotic enumeration of constellations on any fixed surface, with several constraints of face-degree. In particular I have proved conjectures about the universality of the asymptotic counting properties of constellations (and maps) on surfaces.
• A covered map of genus 2.
  A covered map is a map carrying a distinguished unicellular spanning submap (possibly of lower genus than the map itself, as here, where the map has genus 2 but the spanning unicellular submap has genus 1). Covered maps turn out to have very intersting combinatorial and structural properties, as spanning trees have in the planar case. Together with Olivier Bernardi, we introduced and studied covered maps in this paper. In addition to their own properties, covered maps can be used to give new proofs of a priori non-covered-maps-related combinatorial identities.
• A unicellular map of genus 2.
  The vertices in red are the intertwined node of the map. By deleting one of these vertices, one creates a new unicellular map, of lower genus. This paper introduces and uses the notion of interwined node to give new bijection for (almost all) unicellular maps. Applied to labelled unicellular maps, it gives new connections between maps on higher genus surfaces and the ISE random measure.
  Note: my FPSAC'09 paper extends this idea to all unicellular maps, not only almost all (see my main page).