After quickly stating the question I'll tell you about categorification (a bold suggestion of I. Frenkel, that much of math is the Euler characteristic of some "higher math", much like much of algebra is q-algebra at q=1). I'll then define traces and trace groups, which allow Euler characteristics to take values in objects more interesting than merely numbers. Finally I'll introduce the category of matrix factorizations, which is the core of a surpising new method for constructing homological theories from local data.
The ideas to be introduced in my talk (categorifcation, trace groups and matrix factorizations) are all conceptual and foundational and worthy of your time, definitely more than the incomplete (though possibly valid) logic that lead us to our question to Nantel. So assuming some luck, I'll only have time to tell the latter part of the story over coffee after my talk. I hope there's good coffee up there north of 401.
See also: Khovanov and Rozansky's arXiv:math.QA/0401268 and my Khovanov's Homology for Tangles and Cobordisms.