Title: Representing discrete Morse functions with polyhedra



Abstract: Discrete Morse theory is a method of reducing a CW complex to a simpler complex with similar topological properties. Well-known approaches to this task are due to Banchoff, whose process involves embedding a polyhedron in Euclidean space and considering the projections of its vertices onto a straight line, and to Forman, whose process involves finding special functions from the face poset of a complex to the real numbers. In this talk, I will discuss a result by Bloch which gives a relationship between these two methods. In particular, given a discrete Morse function on a CW complex, there exists a corresponding polyhedral embedding of the barycentric subdivision of X such that the discrete Morse function and the projection of the vertices of the polyhedron onto a line give the same critical cells.