TITLE: Applications of quadratic forms
QUESTION [7 upvotes]: It seems that a lot of great mathematicians spent quite a while of their time studying quadratic forms over $\mathbb{Z},\mathbb{Q},\mathbb{Q_p}$ etc. and there is indeed a vast and detailed theory of these. It usually qualifies as part of Number Theory as in Serre's book "A Course In Arithmetic" half of which is devoted to the topic, but it is also claimed to have applications in other areas such as differential topology, finite groups, modular forms (as stated in the preface of the aforementioned book). I would like to have examples of such applications just for general education in order to truly appreciate its universal importance.

REPLY [3 votes]: You mention simple groups. Quadratic forms are a quiet element in the classification of Lie algebras, through Weyl chambers and so on. The short version is that any discussion involving Euclidean spaces that talks about reflections is using a quadratic form, typically the ordinary dot product.  
So, finite simple groups were discovered as integral automorphism groups of integral lattices, meaning other positive quadratic forms over $\mathbb Z.$ This area has always used the language of Lie algebras, Dynkin diagrams, etc. However, the defining example is the Leech lattice, which is a peculiar object. Furthermore, the main calculation taking the Leech lattice to simple groups takes a detour through indefinite forms. 
Three good references are ERROR CORRECTING, LATTICES AND CODES, and any edition of SPLAG.  I am especially fond of section 4.5 in Lattices and Codes by Ebeling. In the preface to the second edition, he points out that section 4.5 is new in that edition. Amazing, as there was no other place that explained what I needed. There is something a little off about availability and price, I think I bought it for about $35.00 from the AMS, but now AMAZON.  I wrote to Prof. Ebeling, he quite liked my application of the section. Known to Conway and Borcherds, of course, but worth publishing as a separate item. We will see.