Algebraic invariants of balanced simplicial complexes

Abstract: Given a simplicial complex which triangulates a certain manifold it is natural to ask what conditions the topology poses on the number of faces in each dimension. Via the Stanley-Reisner correspondence between simplicial complexes and squarefree monomial ideals we can take advantage of tools from commutative algebra, by studying related algebraic invariants such as the Hilbert function and the graded Betti numbers of a certain graded ring. In particular we focus on balanced simplicial complexes, i.e., $(d-1)$-dimensional complexes whose graph is $d$-colorable. After providing the necessary background we present upper bounds on graded Betti numbers of various objects in this family. This is joint work with Martina Juhnke-Kubitzke.