Time permitting, I'll introduce the concept of a projective class on an abelian category $A$ and discuss a generalized existence question: when does the category of chain complexes of objects in $A$ have a model category structure that reflects the relative homological algebra associated to a given projective class?

The key idea is to consider the moduli space of all such topological realizations $X$. Then, building it one stage at a time, the problem can be reduced to an obstruction-theoretic one.

We seek a sort of generalization of this phenomenon. Instead of looking at a cohomology from the eyes of the prime $p$, we consider instead the $p$-completion of our classifying space, $BG^\wedge_p$ (thought of as the $BG$ seen from the eyes of $p$). Martino and Priddy asked the following question: Given two finite groups $G$ and $H$, when do we have a homotopy equivalence of their $p$-completed classifying spaces? They conjectured that this is the case if and only if their "$p$-local data" is the same, and proved the only if implication. Later, the reverse implication was proved by Oliver using the work of Broto-Levi-Oliver (and the Classification of Finite Simple Groups).

The goal of this talk is to explain what is meant by "$p$-local data," which will lead naturally to the work of BLO. Interested parties will finally get to hear what these "saturated fusion systems," "centric linking systems," and "$p$-local finite groups" that I've been jabbering about are, at least as concepts. Rigor will depend on time constraints and my mood.