Organizers: Laura Anderson, Eric Swartz, and Thomas Zaslavsky.
The colored Kruskal-Katona theorem, which extends the Kruskal-Katona theorem, is equivalent to a numerical characterization of the f-vectors of colored simplicial complexes. The underlying theme is the study of initial sets of the reverse lexicographical order. In this talk, I will give a generalization of the colored Kruskal-Katona theorem, explain its relation to the study of initial sets, and discuss its consequences for the f-vectors of colored complexes and balanced complexes of arbitrary type.
Suppose that K and L are compact convex subsets of n-dimensional Euclidean space, and suppose that every (n-1)-dimensional orthogonal projection (that is to say, every shadow) of L onto a subspace contains a translate of the corresponding projection of K to that same subspace. This covering condition does not imply that L contains a translate of K. In fact, we will see that it is even possible for L to have strictly smaller volume! This leads to several questions:
Some of these results arise from joint work with Christina Chen and Tanya Khovanova.