Toward a Galois theory of the integers over the sphere spectrum

Recent work (Ando et al., 2015, 2014; Beardsley, 2017; Hess, 2010; Rognes, 2008) in higher algebra allows the reinterpretation of a classical description (Blumberg, 2010; Cohen et al., 1981; Mahowald, 1977; Mahowald et al., 2001) of the Eilenberg-MacLane spectrum HZ as a Thom spectrum, in terms of a kind of derived Galois theory. This essentially expository talk summarizes some of this work, and suggests an interpretation in terms of configuration spaces and monoidal functors on them, with some analogies to a topological field theory.