Mathematics

In this episode, we will continue to explore the origins of Koszul Duality. We’ll begin again with a reminder of what it means to find a resolution for a module/algebra, and then observe some “coalgebra-like”-behavior on the generators of such a thing. This will lead us to define “coalgebra,” and practice using co-generators and co-relations to build new examples of coalgebras. With these tools, we can make a precise statement regarding the coalgebraic structure inside the resolutions of algebras that we explored at the start.

I will outline the basics of balance theory and sentiment reconstruction and how they relate to cognitively consistent decision-making and data bipartitioning. A few small-scale examples will be calculated to illustrate how this method differs from traditional spectral and random-walk methods. We will then examine a series of clusterability ranking metrics that are able to discern between outcome, intent, and reliability of prediction. Combined, they are able to determine the degree of homophily of the data --- that is, does the data appear to have a human-like desire, or is it synthetic in nature?

The talk will include an assortment of data sets from my recent studies that will hopefully stimulate collaborative interest. Time permitting, I will discuss any open studies, computational bottlenecks, generalizations of the model, and applications to metamaterial construction in spintronics.