The Sphere Packing Problem in Dimension 8 and the Theory of Modular Forms

This thesis is inspired by a paper by Maryna S. Viazovska and studies the connection between modular forms and sphere packings. In particular, we prove that the E8 lattice packing is optimal in the sense that it gives a solution to the sphere packing problem in dimension 8. We start by introducing the problem and putting it in a historical context, we then proceed with a small section in which we introduce the E8 lattice and some general lattice theory. The next two sections built up a lot of theory from different areas of mathematics where we put a particularly strong emphasis on the theory of modular forms. This theory is then used in the following section which is more constructive in nature. All this theory from seemingly independent areas is then tied together in the last section and culminates in a proof, giving a solution to the sphere packing problem in dimension 8.