Organizers: Laura Anderson, Michael Dobbins, and Thomas Zaslavsky.
Consider the collection of all the simplices spanned by some n-point set in Rd. There are several results showing that simplices defined in this way must overlap very much. In this talk I focus on the generalization of these results to 'curvy' simplices.
Specifically, Pach showed that every d+1 sets of points, Q1, ..., Qd+1, in Rd contain linearly-sized subsets Pi in Qi such that all the transversal simplices that they span intersect. In joint work with Alfredo Hubard, we show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size C(log n)1/(d-1). We show that this is tight in dimension 2, for all surfaces other than S2. Surprisingly, the optimal bound for S2 is (log n)1/2. This improves upon results of Barany, Meshulam, Nevo, Tancer.
Given a poset (a partially ordered set), one obtains another poset by considering the collection of intervals of the first, partially ordered by inclusion. (There are various possibilities, depending, for instance, upon whether one considers the empty set as being an "interval.") This construction has found use in the study of convex polytopes and other places. I describe a new method of representation of posets by utilizing certain geometric complexes in Rd having vertices in Zd. The striking feature of this method of representation is that taking the interval poset corresponds to dilation by a factor of 2 of the geometric complex. I explore connections with the integer partitions of powers of 2 into powers of 2.