Inequalities for Polytopes and Permutations, and Homology for Newtonian Coalgebras

The Principle Investigator studies a new class of inequalities on the flag f-vector of convex polytopes. The flag f-vector contains all the enumerative incidence information between the faces of a polytope. Thus to classify the set of all possible flag f-vectors is one of the great open problems in discrete geometry. To date only partial results to this problem have been obtained. In a related problem, the Principle Investigator and his Research Assistant will study the homology groups of Newtonian coalgebras that arise in polytopal theory. These homology groups were recently used to give an algebraic proof of the existence of the cd-index, an important invariant in poset theory. Understanding the homology of these chain complexes will give insight into new applications of Newtonian coalgebras. Polytopes are basic mathematical objects that appear in discrete geometry. The methods to attack these problems are unusually interdisciplinary as they involve insights from both geometry and algebra. The results of this investigation are important as they relate both to the pure branches of mathematics, such as commutative algebra, algebraic geometry and Hopf algebras, and to the applied sciences, including optimization and computer science. Understanding the enumerative information of polytopes will give a theoretical basis for stress and rigidity problems in mechanical engineering, for the reconstruction using sampling problem in computer vision, and for determining the complexity of geometrically-based problems in optimization.