the Sphere Packing Problem in Dimenstion 8 and the Theory of Modular Forms – University of Copenhagen

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the Sphere Packing Problem in Dimenstion 8 and the Theory of Modular Forms

pecialeforsvar ved Kasper Bro Müller

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

Abstract: 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 $E_{8}$ 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 construc-tive 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

 

Vejleder:  Morten S Risager
Censor:    Simon Kristensen, Aarhus Universitet