I will give a survey on my work over the combinatorics at the successor of
a singular cardinal in these two different scenarios:
-the singular cardinal is larger than a strongly compact cardinal $\kappa$
and of cofinality smaller than $\kappa$,
-A strong forcing axiom like Martinīs maximum or the proper forcing axiom
PFA holds.
Under these assumptions I will show that a variety of combinatorial
objects which I call covering matrices are useful tools
to attack many of the classical problems on successor of singular
cardinals, for example we can obtain proofs of Solovay's theorem that the
singular cardinal hypothesis SCH holds above a strongly compact cardinal,
or that PFA implies SCH, or a recent characterization by me and Assaf
Sharon of the structure of the approachability ideal for successors of
singular cardinals of the type described above, etc...
Further informations available at:
www.logic.univie.ac.at/~matteo