Simplicial distributions and polyhedral geometry

Abstract: Simplicial distributions are collections of probability distributions that satisfy certain compatibility conditions that can be encoded topologically using simplicial sets. For a simplicial scenario where the measurement space X and outcome space Y are finitely generated the space sDist(X,Y) of allowed simplicial distributions is a convex set, in fact, a convex polytope. By the Minskowski-Weyl theorem of polytope theory it is well-known that there are two equivalent descriptions of a convex polytope as the intersection of finitely many half-space inequalities (H-representation) or as the convex hull of finitely many extreme points (V-representation). In this talk we detail how one constructs the H-representation of sDist(X,Y) and discuss the conversion to its V-representation, known as the vertex enumeration problem. Time permitting, we will also discuss the Bell polytope, which delineates the boundary between contextual and noncontextual measurement statistics, and is a subpolytope of sDist(X,Y).

(This talk is part of the reading seminar series on the theory and applications of simplicial distributions.)

algebraic topologycategory theory

Audience: researchers in the topic

Bilkent Topology Seminar

Series comments: Contact the organizer to get access to Zoom.

Recordings of talks available at www.youtube.com/channel/UCLrmyGpqxyeVpTcA1b5HcMw/videos