The objects of arithmetic geometry are not manifolds. Some concepts from differential geometry admit analogues in arithmetic, but they are not straightforward. Nevertheless, there is a growing sense that the right way to understand certain Langlands phenomena is to study quantum field theories on these objects. What hope is there of making this thought precise? I will propose the beginnings of a mathematical framework via a general theory of factorization algebras. A new feature is a subtle piece of additional structure on our objects – what I call an _isolability structure_ – that is ordinarily left implicit.

It is a classical fact of rational homotopy theory that the $E_\infty$-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still $E_n$-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of $E_n$-algebras, closely related to Koszul duality for the $E_n$-operads. We will also see that these results are essentially sharp: if the R-valued cochains of $S^n$ have square-zero $E_{n+1}$-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.

 A commensuration of a group G is an isomorphism between finite-index subgroups of G. Equivalence classes of such maps form a group, whose importance first emerged in the work of Margulis on the rigidity and arithmeticity of lattices in semisimple Lie groups. Drawing motivation from this classical setting and from the study of mapping class groups of surfaces, I shall explain why, when N is at least 3, the group of automorphisms of the free group of rank N is its own abstract commensurator. Similar results hold for certain subgroups of Aut(F_N). These results are the outcome of a long-running project with Ric Wade. An important element in the proof is a non-abelian analogue of the Fundamental Theorem of Projective Geometry in which projective subspaces are replaced by the free factors of a free group; this is the content of a long-running project with Mladen Bestvina.