Combinatorics of Triangulations and Hilbert Series

We introduce the basic concepts of Gröbner basis theory and its relations to polytope theory. This will cover very basic parts of [1], the basic chapters from [3] and the first chapters from [11]. In particular, we will define the Gröbner fan, which will play a major role in some of the research problems we will pose later. In this lecture we relate triangulations to Gröbner bases and Hilbert-series. In particular, we study f-vectors of simplicial complexes and its Stanley-Reisner ideals. The basic material can be found in the book [4] and the book [11]. 3. LECTURE: GR¨OBNER BASES FOR HIBI-RINGS AND OTHER EHRHART-RINGS In this lecture we exhibit the connections between the Neggers-Stanley conjecture, Hibi-rings and Gröbner basis. Then we present the results that show how to generalize this to arbitrary Ehrhart-rings. The lecture is based on results from [7], [2] and [5]. 4. LECTURE: SPECIAL GR¨OBNER BASES FOR DETERMINANTAL AND PFAFFIAN IDEALS In this lecture we present special Gröbner bases for Determinantal and Pfaffian ideals that allow strong conclusions on the Hilbert-series. In particular, we relate those Gröbner bases to polytope theory and classical polytopes such as the associahedron and the cyclo-hedron. This lecture is based on [6], [10] and background from [9] and [8].