Event Details

John Haslegrave (lecturer). Planar percolation: from perimeters to probability. Facundo Canale (PhD Student). Infinite Galois Groups

Speaker 1: John Haslegrave (lecturer)

Planar percolation: from perimeters to probability

Abstract: Percolation was introduced in statistical physics as a way to model flow in a disordered medium, and comes in two main flavours: “site” and “bond”. As the parameters vary, there is a critical point when long-range flow becomes possible. Paradoxically, we now know a lot about the behaviour at this point, even though we can only say where it is in a few special cases. I will introduce the model in some familiar two-dimensional settings, and then discuss recent progress on some conjectures of Benjamini and Schramm about how the critical point can change when the underlying structure is less symmetric.

Speaker 2: Facundo Canale (PhD Student)

Infinite Galois Groups

Abstract: The study of the behaviour of the roots of polynomials was consolidated in the 1700s by Galois (and others) in what is now known as “Galois theory”.

Galois’ work led to the first axiomatisation of a group and consequently to (arguably) the birth of group theory. In modern language, given a field k and a field extension K/k, the Galois group of K/k is the group of automorphisms of K that fix k.

The groups Galois studied were always obtained from field extensions of finite degree and hence of finite order, but his theory was later extended to field extensions of infinite degree and their corresponding Galois groups of infinite order. These are known as profinite groups, of which finite groups are a special case.

It turns out that giving an appropriate topology to these groups one can obtain a reasonable “Galois Correspondence” for these groups by only considering closed subgroups.

In this talk we will introduce profinite groups as inverse limits of finite groups and see how they relate to Galois groups of field extensions. We will give some examples of Galois Groups that are known and some that remain open.