Egde ideals of graphs play a prominent role in combinatorial commutative algebra. The homological properties of these ideals and their powers reflect very well the combinatorics of the underlying graph. In this lectures we not only consider edge ideals of graphs but also those of hypergraphs. We aim at presenting the known facts about the powers of these ideals. This includes the description of their associated prime ideals and the depth function.

The study of face numbers of simplicial complexes is a classical problem in algebraic combinatorics. We will provide an overview of results and methods used in this area, including classical results (e.g., the g-theorem for simplicial polytopes) as well as more recent ones for manifolds with (and without) boundary (e.g., the study of the Sigma module).

Local cohomology is difficult to compute explicitly. One can reduce to a simple set-up as in the Hochster's formula for the local cohomology of a Stanley-Reisner ring. The poset defined by the primary decomposition of the defining ideal provides the needed combinatorial information. These ideas can be extended in several directions, always with the above poset as the combinatorial object where to look. In these lectures we shall review some of these constructions, which often involve the explicit computation of the derived functors of the direct and inverse limits over a finite poset.

In these lectures we introduce a class of binomial ideals attached to graphs, called "binomial edge ideals", and we study various algebraic properties and invariants of them. We try to understand those properties and invariants via the combinatorial properties  of the underlying graph. Finally, we present some open problems in this area.    

In these lectures we discuss lattice polytopes and associated rings. The latter are toric algebras. The interplay between combinatorial properties of the polytopes and the corresponding algebraic ones of the algebras is an active area of research in combinatorial commutative algebra. We discuss examples of such results and useful methods to study these objects. Moreover, some lattice polytopes of interest are considered to which we apply the theory.

We discuss two aspects of monomial ideals. Many algebras are expected to have the weak Lefschetz property. However, establishing this property is often very challenging. For monomial ideals there are many connections to combinatorial problems. The Waldschmidt constant and the resurgence are two invariants of an ideal that arise in asymptotic studies of symbolic powers of ideals. Determining these invariants is typically difficult, even for a monomial ideal.

The Betti table of the minimal free resolution of a Stanley-Reisner ideal of a simplicial complex is by Hochster's formula governed by the simplicial homology of induced subcomplexes. Therefore extremal behavior of the Betti table (i.e., the resolution is linear for A steps and has regularity B) is reflected in extremal behavior of the simplicial complexes. Thus the existence of such simplicial complexes implies that existence of squarefree monomial ideals with a certain Betti table. In the lecture we give examples of questions on simplicial complexes that arise from commutative algebra in that way. Then we provide methods how to prove the existence of such simplicial complexes, e.g. through explicit geometric constructions or probabilistic arguments.