The control polygon of a Bezier curve is well-defined and meaningful---there
is a sequence of weights under which the limiting position of the curve is the
control polygon. For a Bezier surface patch, there are many possible polyhedral
control structures, and none are canonical. We propose a not necessarily
polyhedral control structure for surface patches, regular control surfaces,
which are certain C^0 spline surfaces.

An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes.

We present a new continuation algorithm to find all nondegenerate real
solutions to a system of polynomial equations. Unlike homotopy methods, it is
not based on a deformation of the system; instead, it traces real curves
connecting the solutions of one system of equations to those of another,
eventually leading to the desired real solutions. It also differs from homotopy
methods in that it follows only real paths and computes no complex solutions of
the original equations.

The multiplihedra {M_n} form a family of polytopes originating in the study
of higher categories and homotopy theory. While the multiplihedra may be
unfamiliar to the algebraic combinatorics community, it is nestled between two
families of polytopes that certainly are not: the permutahedra {S_n} and
associahedra {Y_n}. The maps between these families reveal several new Hopf
structures on tree-like objects nestled between the Malvenuto-Reutenauer (MR)
Hopf algebra of permutations and the Loday-Ronco (LR) Hopf algebra of planar
binary trees.