Algebraic and combinatorial aspects of polytopes and domino tilings

Two classical objects of study in combinatorics are polytopes and domino tilings. In the 1990s William Thurston proved that the set of domino tilings of a simply- connected region is connected by flips. The first part of this thesis will introduce definitions associated to the underlying grid graph of a collection of domino tilings. We will then provide an analogous proof of Thurston’s result by using the language of toric ideals. The second half of this thesis, a joint project with Anastasia Chavez, focuses on d-dimensional simplicial polytopes. In particular, we introduce the associated face vector i.e. the f-vector of a d-dimensional polytope. In conclusion, we prove a correspondence between the maximal linearly independent subsets of the f-vectors of simplicial polytopes and the Catalan numbers.