Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents

This proposal is about combinatorial algebra, with a geometrical flavor. Together with students and collaborators I have been developing a diverse family of graded algebras and coalgebras, modules and comodules, often with Hopf and differential structures. Their common feature is that all are based upon recursive sequences of convex polytopes. The key point of interest in each case is to see how the algebraic structure reflects the combinatorial structure, and vice versa. We are building upon the foundations laid by many other researchers, especially Gian-Carlo Rota, who most clearly saw the strength of this approach. The historical examples of Hopf algebras SSym and QSym, the Malvenuto-Reutenauer Hopf algebra and the quasisymmetric functions, can be defined using graded bases of permutations and boolean subsets respectively. Loday and Ronco used the fact that certain binary trees can represent both sorts of combinatorial objects to discover the Hopf algebra YSym lying between them. Chapoton capitalized on the fact that the three graded bases could actually be described as the vertex sets of polytope sequences, and defined larger algebras on the faces of the permutohedra, associahedra and cubes. The polytope sequences we study include those familiar examples as well as newer families such as the graph multiplihedra and composihedra. Simultaneously with our study of Hopf algebras we

In 2008 several researchers made exciting advances in the application of algebra and combinatorics to particle physics. The crucial mathematical ingredients included Hopf Algebras and subalgebras, in regard to the specific physical principle of renormalization in quantum electrodynamics and chromodynamics. Renormalization refers to the addition of counterterms to a sequence of divergent integrals for probability