Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Description

Let A be a finite point set in the real affine space of dimension d. For example, the vertex set of a d-dimensional polytope. A triangulation of A is any family of simplices with vertex set contained in A and which form a geometric simplicial complex covering the convex hull of A.

The "set" of all triangulations of $A$ can be thought as a "space" in two related ways:

- There is a natural concept of local transformation between triangulations called geometric bistellar flip. This defines a GRAPH OF TRIANGULATIONS OF A whose nodes are the triangulations and whose arcs are the bistellar flips between them.

- If we generalize triangulations allowing polyhedral complexes instead of simplicial ones, we have a POSET OF POLYHEDRAL SUBDIVISIONS OF A, whose order relation is "refinement". Triangulations are the minimal elements in the poset. As with any poset, we can associate to this poset the abstract simplicial complex whose simplices are the finite chains (totally ordered subsets) in the poset.

The topology of these two objects is related, for example, to properties of the toric variety defined by A (if A consists of integer points). We will discuss these relations and mention recent results, which include the construction of a point set whose space of triangulations is disconnected