Many operations and equivalences in mathematics appear as a kind of Fourier transform. By Fourier transform, I mean the following:

Let $ X $ and $ Y $ be two objects of a certain category with products, and consider correspondence $ X leftarrow X times Y to Y $. If we have an object (think of the sheaf, the function, the space, etc.) $ mathcal {P} $ more than $ X times Y $ and another, let's say $ mathcal {F} $ more than $ X $, assuming the existence of proper push and pull operations, we can get another object on $ Y $ pulling $ mathcal {F} $ back to product, tensor with $ mathcal {P} $, then pushing towards $ Y $.

The standard example is the Fourier transform of functions on a locally compact abelian group $ G $ (for example. $ mathbb {R} $). In that case, $ Y $ is the Pontryagin dual of $ G $, $ mathcal {P} $ is the exponential function on the product, and the push and pull are given by integration and precomposition, respectively.

We also have the Fourier-Mukai functors for coherent showers in algebraic geometry which provide the equivalence of coherent showers on double abelian varieties. In fact, almost all of the interesting functors between coherent bundles on fairly pleasant varieties are examples of Fourier-Mukai transforms. A variant of this example also provides the geometric Langlands correspondence

$$ D (Bun_T (C)) simeq QCoh (LocSys_1 (C)) $$

for a torus $ T $ and a curve $ C $. In fact, the Langlands geometric correspondence for general reducing groups also seems to come from such a transformation.

Speak $ SYZ $ guess, two Calabi-Yau mirror collectors $ X $ and $ Y $ are Lagrangian double-core fibrations. As such, conjectured equivalence

$$ D (Coh (X)) simeq Fuk (Y) $$

is obtained morally by applying a Fourier-Mukai transform which ignites coherent sheaves $ X $ in Lagrangiens in $ Y $.

To make things more mysterious, many of these examples are the result of the existence of a perfect match. For example, the bundle of Poincaré lines which provides the equivalence of coherent sheaves on double abelian varieties $ A $ and $ A ^ * $ results from the perfect combination

$$ A times A ^ * à B mathbb {G} _m. $$

Likewise, the Langlands geometric correspondence for the toroids, as well as the GLC for the Hitchin system, arise in a way from the self-duality of the Picard stack of the underlying curve. These examples seem to show that the non-degenerate quadratic forms seem to be fundamental in a very deep sense (for example perhaps even the Poincaré duality could be considered as a Fourier transform).

I don't have a specific question, but I would like to know why we should expect Fourier transformations to be so fundamental. These transformations are also found in physics as well as in many other "real world" situations of which I am even less qualified than my examples above. Nevertheless, I have the feeling that something deep is going on here and I would like an explanation, even a philosophical one, on why this model seems to appear everywhere.