Airy equation: a topological approach of its stokes phenomenon

In this thesis, we present different equivalent ways of studying the Stokes phenomenon of the Airy equation. Since it has two entire solutions that can be written as a Fourier transform of an exponential function, in the first chapters we introduce the notions of enhanced sheaves and of their Fourier-Sato transform. For the Airy case this is a compact support first homology group of a specific region, that has as generators some cycles that we can explicitly find by basic Morse theory. In conclusion, these cycles give an isomorphism between the previous Fourier-Sato transform and a direct sum of exponential enhanced sheaves that correspond to the asymptotic behavior of the solutions of the Airy equation.