Béranger Seguin

Since Antiquity, mathematicians have been trying to find rational
solutions to systems of polynomials equations (sometimes in a less
precise language). Such questions are called Diophantine equations.
During the 19th century, it became clear that the key to solving these
equations lay in the study of fields and especially in a central tool:
the Galois group, which measures how "symmetric" a field is. A natural
yet unsolved question is the *inverse Galois problem*: can every finite
group be realized as the Galois group of a finite extension of the field
of rational numbers? In this talk, we will tell the story of how our
viewpoint on this problem has shifted in the last century. We will first
see that the problem is actually related to geometry: are there certain
covers of the projective line which are defined by equations with
rational coefficients? Then, we will see that covers may themselves be
seen as points on a bigger geometrical space: the Hurwitz space. The
inverse Galois problem is then asking about the existence of rational
points on this space: this is itself a Diophantine equation! We will
mention some of the achievements obtained using this paradigm, for
example the realization of the Monster group as a Galois group.