Bhargava and the speaker introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz-Mazur. In this talk, we use Galois closures to construct new components of Hilbert schemes of points, which are fundamental objects in algebraic geometry whose component structure is largely mysterious. We answer a 35 year old open problem posed by Iarrobino by constructing an infinite family of low dimensional components. This talk is based on joint work with Andrew Staal.

This talk aims to present the results obtained by Oussama Hamza, during his PhD studies, and his collaborators: Christian Maire, Jan Minac and Nguyen Duy Tân.

Their work precisely focuses on realisation of pro-p Galois groups over some fields with specific properties for a fixed prime p: especially filtrations and cohomology. Hamza was particularly interested on Number and Pythagorean fields.

This talk will mostly deal with the last results obtained by Hamza and his collaborators on Formally real Pythagorean fields of finite type (RPF). For this purpose, they introduced a class of pro-2 groups, which is called $\Delta$-RAAGs, and studied some of their filtrations. Using previous work of Minac and Spira, Hamza and his collaborators showed that every pro-2 Absolute Galois group of a RPF is $\Delta$-RAAG. Conversely if a group is $\Delta$-RAAG and a pro-2 Absolute Galois group, then the underlying field is necessarily RPF. This gives us a new criterion to detect Absolute Galois groups.

Finally, we also show that the pro-2 Absolute Galois group of a RPF satisfy the Kernel unipotent conjecture jointly introduced, by Minac and Tân, with the Massey vanishing conjecture, which attracted a lot of interest.