Algebraic and enumerative combinatorics seminar - Ian George-Enumerating Convex Sets in Posets

Abstract: Causal Set Theory (CST) is a theory of quantum gravity where spacetime is taken to be a locally finite poset, called a causal set.  A central problem in CST is to determine physically relevant properties from the purely combinatorial information of the causal set.  In 2014 Glaser and Surya demonstrated that the distribution of interval sizes of a causal set sprinkled into a region of Minkowski space contains information about the dimension of the underlying spacetime.  In 2026 Surya showed that this distribution can be used to define a “closeness” function on causal sets that distinguishes by dimension and global topology.  In this talk we present work motivated by these results which investigates the more general notion of convex sets, instead of intervals, of a poset.  First, we will introduce a generating polynomial for convex sets in a finite poset and explore some of its properties.  We will then show that this polynomial is a complete invariant for the family of series-parallel posets.  Lastly, we discuss early results on the utility of this polynomial in CST.  The pre-seminar will introduce relevant background on CST.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm.