There are many examples in algebraic geometry in which complicated geometric or algebraic problems can be transformed into purely combinatorial problems. The most prominent example is probably given by toric varieties — a certain class of varieties that can be described purely by combinatorial data, e.g. by giving a convex polytope in an integral lattice. As a consequence, most questions about ...

Coxeter complexes are simplicial polytopes derived from certain reflection groups. The type An−1 Coxeter complex is derived from the symmetric group Sn and admits an elegant combinatorial description. Its faces are identified with set compositions and certain combinatorial properties of these compositions can be explained geometrically. Aguiar and Mahajan used this geometry to define a product ...

Abstract This survey is about combinatorial objects related to reflection groups and their applications in representation theory arithmetic geometry. Coxeter folded galleries complexes are introduced detail illustrated by examples. Further it explained how they relate retractions Bruhat-Tits buildings the geometry of affine flag varieties Grassmannians. The goal make these topics accessible a w...

We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry a...

Several combinatorial aspects of triangulations and their generalizations are studied in this thesis. A triangulation of a point con guration and a d-dimensional polyhedron whose vertices are among the points is a decomposition of the polyhedron using d-simplices with vertices among the points. The two main elds triangulations appear are combinatorial geometry in mathematics and computational g...

This paper proposes a comprehensive approach to computing a navigation graph for an indoor space. It focuses on a single floor, but the work is easily extensible to multi-level spaces. The approach proceeds by using a formal model, based on the combinatorial map but enhanced with geometric and semantic information. The process is almost fully automatic, taking as input the building plans provid...

Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of p-angulations to arbitrary dimensions. We prove the existence of a bijection between any such family and some colored combinatorial maps which we call stuffed Walsh maps. Those maps gener...

We present a simple iterative strategy for measuring the connection strength between a pair of vertices in a graph. The method is attractive in that it has a linear complexity and can be easily parallelized. Based on an analysis of the convergence property, we propose a mutually reinforcing model to explain the intuition behind the strategy. The practical effectiveness of this measure is demons...