Tam, Ricci Yik Chi2015-01-202015-01-202015-01-202015-01-16http://hdl.handle.net/10012/9105Ehrhart theory studies the behaviour of lattice points contained in dilates of lattice polytopes. We provide an introduction to Ehrhart theory. In particular, we prove Ehrhart's Theorem, Stanley Non-negativity, and Ehrhart-Macdonald Reciprocity via lattice triangulations. We also introduce the algebra $\mathscr{P}(\mathbb{R}^d)$ spanned by indicator functions of polyhedra, and valuations (linear functions) on $\mathscr{P}(\mathbb{R}^d)$. Through this, we derive Brion's Theorem, which gives an alternate proof of Ehrhart's Theorem. The proof of Brion's Theorem makes use of decomposing the lattice polytope in $\mathscr{P}(\mathbb{R}^d)$ into support cones and other polyhedra. More generally, Betke and Kneser proved that every lattice polytope in $\mathscr{P}(\mathbb{R}^d)$ (or the sub-algebra $\mathscr{P}(\mathbb{Z}^d)$, spanned by lattice polytopes) admits a unimodular decomposition; it can be expressed as a formal sum of unimodular simplices. We give a new streamlined proof of this result, as well as some applications and consequences.enPolyhedraEhrhart TheoryPolytope DecompositionValuationsPolytope AlgebraPolyhedral SubdivisionLattice PolytopesMaster ThesisCombinatorics and Optimization