In April this year I’ll be giving a talk, with the title Candidate Betti numbers for the linear homology of convex polytopes. at the Université Libre de Bruxelles. Here are Candidate Betti numbers for linear homology slides that I originally prepared.  And here’s the Finding linear homology slides that I’ll actually use for the seminar. The second set is short, and gets to the main definitions more quickly.  The first set puts the material in context, but is much too much for a single talk. [Second set of slides added April 1, 2013.]

Here’s the abstract:

The middle perversity intersection homology (mpih) Betti numbers of the toric variety associated with a convex polytope are linear functions of the flag vector of the convex polytope.
In this talk I define similar linear functions, which I hope are the Betti numbers for a not yet defined homology theory. This linear homology theory should exist wherever mpih does.
Such homology would prove that these candidate Betti numbers are actual Betti numbers, and so non-negative on all convex polytopes (being the dimension of a vector space).