We will define piecewise polynomial ring over a fan and discuss some properties of this ring. Then we will show that the equivariant cohomology of a toric variety over a fan $\Sigma$ (with vanishing odd cohomology) is given by piecewise polynomials over $\Sigma$.

This is the second part.

Discrete homotopy theory is a way of analyzing graphs with topological ideas. Among other discrete analogs of familiar concepts, it is possible to define discrete homotopy groups and discrete homology groups of a graph which give insight into the connectivity of the graph. In the talk, an introduction to the emerging field will be given. Additionally, a notion of Cat(0) graphs will be introduced, whose theory resembles the theory of Cat(0) cubical complexes. The Cartan-Hadamard theorem for cubical complexes implies that Cat(0) cubical complexes are contractible; a combinatorial analog of this theorem for graphs will be presented. As an application, the discrete homotopy types of Cayley graphs of Coxeter groups will be determined.

A framed polytope is the convex closure of a finite set of points in Euclidean space together with an ordered linear basis. An n-category is a category that is enriched in the category of (n-1)-categories. Although these concepts may initially appear to be distant peaks in the mathematical landscape, there exists a trail connecting them, blazed in the 90's by Kapranov and Voevodsky. We will traverse this path, widening and improving it as we address some of their conjectures along the way. If time permits, using a special embedding of the combinatorial simplex, we will connect this trail to the one ascending Mount Steenrod. This connection will enable us to express the combinatorics of cup-i products in convex geometric terms, dual to those introduced earlier in the talk to define the nerve of higher categories.