Oliver Lindström: Logarithmic Geometry and the S^1-framed Kontsevich Operad

Abstract.

In this thesis, I define, for each positive integer \(d\), an operad in the category of schemes over some base field \(k\), whose objects are the moduli spaces of stable \(n\)-pointed rooted trees of \(d\)-dimensional projective spaces, \(T_{d,n}\). I then define log structures on these spaces and extend the morphisms of this operad to define an operad of log schemes without unit. Finally, I show that the Kato-Nakayama analytification of this non-unital operad is isomorphic to the operadic semidirect product \(K_{2d} \rtimes S^1\) of the Kontsevich operad (without unit) in dimension \(2d\) and the \(S^1\) topological group.