The aforementioned manuscript substantially extends, both conceptually and computationally, our knowledge of the sheaf cohomology groups of the ordinals. These form what set theorists might term a graded family of incompactness principles exhibiting a mix of ZFC and ZFC-independent behaviors; as such, they form important invariants of the set theoretic universe itself. A range of new techniques for their evaluation were developed in the course of the fellowship.
The work on manifold classification consisted, first of all, in the provision of a sufficiently general, or modular, approach to parametrizing manifolds (of a variety of types) as standard Borel spaces. Within this framework, a number of complexity computations were recorded (for hyperbolic manifolds, for topological 2-manifolds, for finite-type hyperbolic 3-manifolds, etc.); at least as significant, though, are the problems which this framework renders newly accessible, i.e. the field which this work opens up.
Our work on condensed mathematics involved, on one front, an analysis of its constructions in terms of set theoretic forcing; we showed that the condensation of nice topological spaces X are, in essence, nothing other than “an organized presentation of all forcing names for elements of X”. On another front, we showed that infinitary combinatorics closely related to the cohomology computations referenced above carry implications for the structure of condensed categories; this was by way of an analysis of those combinatorics’ bearing on multiple derived functor computations.
In our work on the category theory of Polish spaces and groups, we analyzed the derived category of locally compact Polish abelian groups, identifying the injective and projective objects of its heart.
