Polytopes generalize objects such as cubes, tetrahedra, and icosahedra to arbitrary dimensions and they naturally appear in different areas of mathematics and mathematical physics. Lattice polytopes, i.e., polytopes whose vertices have integral coordinates, form the unifying thread in the research project at hand.
First, using these lattice polytopes, we want to examine the macro-scale behavior of crystalline lattices coming from the micro-scale interactions between neighbors. Using lattice polytopes, we want to find an efficient way of determining this behavior.
Second, we propose to use lattice polytopes to examine symmetries related to hyperplane arrangements.
Third, we want to understand some properties of lattice polytopes related to partially ordered sets, i.e., sets where some elements are comparable.
Lastly, we propose to create a special family of lattice polytopes lacking a certain property, thus leading to a better understanding of lattice polytopes.