  In this paper we describe a number of extensions to Razborov's semidefinite
flag algebra method. We will begin by showing how to apply the method to
significantly improve the upper bounds of edge and vertex Tur\'an density type
results for hypercubes. We will then introduce an improvement to the method
which can be applied in a more general setting, notably to 3-uniform
hypergraphs, to get a new upper bound of 0.5615 for $\pi(K_4^3)$.
  For hypercubes we improve Thomason and Wagner's result on the upper bound of
the edge Tur\'an density of a 4-cycle free subcube to 0.60318 and Chung's
result on forbidding 6-cycles to 0.36577. We also show that the upper bound of
the vertex Tur\'an density of $\mc{Q}_3$ can be improved to 0.76900, and that
the vertex Tur\'an density of $\mc{Q}_3$ with one vertex removed is precisely
2/3.
