Final Report Summary - FEN (Face enumeration for spheres,<br/>balanced skeleta of polytopes,<br/>and arrangements) We study the face enumeration of triangulated spheres and their relatives. This includes the family of all simplicial spheres, flag spheres, subfamilies interpolating between these two, balanced skeleta in polytopes and levels in hyperplane arrangements.The proposed research lies at the intersection of enumerative combinatorics, commutative algebra, Euclidean geometry and convexity. During the last few decades, several textbooks have been written on the connections between these fields, and they have been extensively studied.Among the main problems we address are the well known g-conjecture for simplicial spheres and the Charney-Davis conjecture for flag simplicial spheres. We plan to address them from a new perspective, by considering a filtration defined by the families of simplicial d-dimensional spheres without missing-faces of dimension larger than a fixed threshold i (i=1 is the flag case, i=d+1 is the case of all spheres). We look for an algebraic structures for these subfamilies of simplicial spheres that will provide sharp lower bounds on their face numbers. We note that the classical Stanley-Reisner ring is not the right object for the intermediate cases 1Besides tools from commutative algebra, tools from framework rigidity, and variants of them, will be used to address related problems on face numbers for levels in hyperplane arrangements (generalizing McMullen's upper bound theorems for simplicial polytopes) and for balanced skeleta of polytopes, for example to Adin's conjecture on face numbers of cubical polytopes.