Treffer: HOMOLOGICAL ALGEBRA AND PROBLEMS IN COMBINATORICS AND GEOMETRY

This dissertation uses methods from homological algebra and computational commu-tative algebra to study four problems: a conjecture about the dimension of the space of piecewise polynomial functions which are C r on a triangulation ∆ ⊆ R 2; a criteria on the k−formality of hyperplane arrangements and an application to graphic ar-rangements; generalizations of results in hyperplane arrangements to configurations of smooth rational curves in P 2; and a problem in algebraic coding theory on evalua-tion codes and their minimal distance. The methods used are mostly Hilbert function computations of certain graded modules, the study of syzygies, the homology of chain complexes, and geometric results such as the Cayley-Bacharach theorem. The dissertation is made up of 5 chapters. The first chapter is an introduction where the basics of commutative and homological algebra, simplicial complexes and hyperplane arrangements are introduced. The remaining four chapters are each ded-icated to one of the problems listed above. To note here that these chapters have two parts, a first part introducing the problem, and a second part consisting of the