Beyond Grothendieck's conjecture on Galois groups and arithmetic fundamental groups

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The pioneering work of Galois, on the solvability by radicals of polynomials equations, established group theory at the heart of modern mathematics. The philosophy inspired by the work of Galois is that given a set of algebraic polynomial equations, various natural symmetries of the set of solutions of these equations encode some important intrinsic properties of the equations. This philosophy was empowered by the far reaching vision of Grothendieck in the second half of the last century. Grothendieck conjectured that finitely generated fields can be reconstructed quite naturally from certain groups arising as symmetry groups of solutions of certain algebraic polynomial equations; the so-called absolute Galois groups. More generally a certain class of algebraic varieties; these are mathematical objects defined by algebraic polynomial equations, can be reconstructed quite naturally from arithmetic fundamental groups arising as symmetry groups of solutions of certain algebraic polynomial equations. The vision of Grothendieck concretised in the theorems of Neukirch, Uchida, Pop, Tamagawa, and Mochizuki. They established the main foundational results of the so-called anabelian geometry and its birational version. Unfortunately, Galois groups of finitely generated fields, and likewise arithmetic fundamental groups, are still very mysterious objects. A full and explicit understanding of these objects seems to be out of reach in the foreseeable future. Class field theory provides an explicit description of some rather small portion of Galois groups and arithmetic fundamental groups: the abelian quotients. The main objective of this proposal is to establish new results in this area of mathematical research, whereby one can naturally reconstruct finitely generated fields, as well as certain algebraic varieties, from some quotients of Galois groups and arithmetic fundamental groups; the so-called m-step solvable quotients, which are better understood. These quotients are built up successively, step by step, starting from abelian quotients which are rather well understood by class field theory. Such results would be a substantial sharpening of the foundational results of the theory, and would pave the way to a more explicit, and applicable, Galois theory of finitely generated fields and algebraic varieties.