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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.

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If $X$ is an orientable manifold of dimension $n = 4k$, then the Poincare Duality pairing induces a unimodular integral symmetric bilinear form on $H_{2k}(X,\mathbb{Z})$. This form turns out to be a deep and useful invariant of $X$. For instance, I quote (or rather, closely paraphrase) the following result taken from Milnor and Husemoller's Symmetric Bilinear Forms:

Theorem: Let $M,M'$ be closed, oriented simply connected $4$-dimensional manifolds. The following are equivalent:
(i) There is an orientation-preserving homotopy equivalence from $M$ to $M'$.
(ii) The symmetric inner product space $H_2(M,\mathbb{Z})$ is isomorphic to $H_2(M',\mathbb{Z})$.

There are other amazing results along these lines: see e.g. Rokhlin's Theorem.

(In fact, I want to say that something about the intersection form defined above gives an obstruction for a topological manifold to admit a smooth structure, which is about as deep as it gets in this subject. Unfortunately I am not remembering the actual statement now: perhaps someone can remind me.)

I have always assumed that geometer-topologist extraordinaire John Milnor got interested enough in symmetric bilinear forms to write an entire book about them because of this application to manifold theory, although I looked just now and didn't find any explicit statement along these lines in the book.

Note finally that Chapter V of Milnor-Husemoller is called "Some Examples" and the entire discussion above concerns $\S V.1$: Homology Theory of Manifolds. I suspect that one could write two more good answers using $\S$ V.2: Rings of Smooth Real Valued Functions and $\S$ V.3: The Discriminant of a Field Extension.

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Offering a very different application: the Gold sequences (used in e.g. GPS-navigation to track changes to the distance of one of the satellites) can be viewed and analyzed using the fact they are gotten by evaluating certain quadratic forms (depending on the id-number of the satellite) defined on a 10-dimensional space over the field of two elements.

Several other widely used synchronization/scrambling sequences (but not all) are gotten in the same way.

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Let $A$ be a finite-dimensional algebra over a field $k$. Then there is a distinguished linear functional $\text{tr}(L_a) : A \to k$ given by computing the trace of the left multiplication operator $L_a : x \mapsto ax$. This induces a distinguished symmetric bilinear form $\langle a, b \rangle = \text{tr}(L_a L_b)$, the trace form. This is a useful tool for studying $A$.

Theorem: $A$ is a semisimple algebra if and only if the trace form is nondegenerate.

In this case, $A$ is equipped with the structure of a Frobenius algebra, so it can be used to define 2-dimensional topological quantum field theories.

Theorem: Similarly, let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$ of characteristic $0$. The trace form in this context is called the Killing form, and by the Cartan criterion $\mathfrak{g}$ is semisimple if and only if the Killing form is nondegenerate.

When $k \to A$ is an extension of number fields, the trace form can be used to define the discriminant of a number field, which alone has many applications in algebraic number theory.

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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.

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