Combinatorial Methods in Topology and Algebra Combinatorics plays a prominent role in contemporary mathematics due to the vibrant development it has experienced in the last two decades and its many interactions with other subjects. This book arises from the INdAM conference CoMeTA 2013 Combinatorial Methods in Topology and Algebra  which was held in Cortona in September 2013. The event brought together emerging and leading researchers at the crossroads of Combinatorics Topology and Algebra with a particular focus on new trends in subjects such as: hyperplane arrangements; discrete geometry and combinatorial topology; polytope theory and triangulations of manifolds; combinatorial algebraic geometry and commutative algebra; algebraic combinatorics; and combinatorial representation theory. The book is divided into two parts. The first expands on the topics discussed at the conference by providing additional background and explanations while the second presents original contributions on new trends in the topics addressed by the conference 1 Introduction 2 Part I 3 Extremal graph theory and face numbers of flag triangulations of manifolds 4 Combinatorial stratifications and minimality of 2 arrangements 5 Rota’s conjecture the missing axiom and prime cycles in toric varieties 6 Random triangular groups 7 Generalized involution models of projective reflection groups 8 Universal Gröbner bases for maximal minors of matrices of linear forms 9 Milnor fibrations of arrangements 10 Decompositions of Betti diagrams of powers of monomial ideals 11 Matroids over a ring 12 A Minimal Irreducible Triangulation of S3 13 Tropical Oriented Matroids 14 A combinatorial classification of Buchsbaum simplicial posets 15 Differences Between Nonnegative Polynomials and Sums of Squares 16 Interpolation box splines and lattice points in polytopes 17 Root polytopes of crystallographic root systems 18 Product formulas for volumes of flow polytopes 19 On the Topology of the Cambrian Semilattices 20 cd index for CW posets 21 Bipartite Rigidity 22 Balanced manifolds and pseudomanifolds 23 Some combinatorial constructions and relations with Artin groups 24 Deterministic Abelian Sandpile and square triangle tilings 25 A special feature of quadratic monomial ideals 26 Resonant bands local systems and Milnor fibers of real line arrangements 27 Full research papers 28 On Highly Regular Embeddings 29 Positive sum systems 30 The Sn+1 action on spherical models and supermaximal models of tipe An 1 31 h vectors of matroid complexes