DescriptionWhich positive integers can be written as a sum of three squares? Of four squares? And moreover, in how many ways can we write an integer as a sum of three squares or four squares? It turns out that these questions, and even more general forms of these questions, can be studied using the theory of Quadratic Forms.The main aim of this project is to learn the basic theory of Quadratic Forms. We will start by reading the first part of the beautiful book of Serre "A Course in Arithmetic" [3] combined with an interesting exposition to the subject by John Conway (The Sensual Quadratic Form [2]). Then, there are various interesting directions that the project may take. Below we list a few of them:
Possible Directions:Classification of Quadratic Forms : We will develop various notions of equivalence between Quadratic Forms, and then criteria for determining whether two Quadratic Forms are equivalent.Quadratic Forms and Theta Series: The main aim of this direction is to study the link of the theory of Quadratic Forms with analysis. Namely we will see that there exists some complex functions, called Theta Series, which encode information about Quadratic Form. Quadratic Forms and Sphere Packing: Quadratic Forms are closely related to Lattices, and these in turn to the so-called Sphere Packing Problem. The main aim of this direction is to investigate this relation.
ResourcesThe following books will be used as references
[2] J.H. Conway, The Sensual Quadratic Form, The Carus Mathematical Monographs, Number 26, 1997. [3] J.-P.Serre, A Course in Arithmetic , Graduate Texts in Mathematics 7, Springer
Prerequisites
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