Numerical validation and applications

Numerical validation is a small branch of maths, but its small size not withstanding, it's not lacking in charm, or in interesting results!
In this talk, I plan on giving a brief summary of the history of validated numerics, in particular in dynamical systems, and the most popular techniques, then I will present my own research: the validation of bifurcations.
We will talk about periodic orbits, Hopf bifurcations and codimension 2 bifurcations, as well as of theoretical numerical analysis and applications in machine learning.