An important goal of modern Galois theory is to characterize which profinite groups can arise as absolute Galois groups of fields. Following the 2011 proof of the Bloch-Kato conjecture, which established the quadratic nature of the \(\mathbb{F}_p\)-cohomology of certain maximal pro-\(p\) Galois groups, several related conjectures have emerged, aimed at refining our understanding of these cohomology rings.
A key tool in this context is the linearization process, which associates a restricted Lie algebra to any pro-\(p\) group. The cohomology of such a Lie algebra is intimately connected to the cohomology of the group itself. In this talk, we will explore the Lie algebraic formulations of several Galois-theoretic conjectures, presenting evidence that either supports or challenges their validity.