Complaints of the foreign Congress of Mathematicians 2014. Seoul ICM 2014.

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We say that A is amenable if it satisfies all of the following properties: (1) (Invariance under weak isomorphisms) If f : X → C is in A and g : Y → C is weakly isomorphic to f , then g ∈ A. (2) (Closure under restrictions and unions) If X ⊆ P(s) is downward-closed and f : X → C is a functor, then f ∈ A if and only if for every u ∈ X, we have that f P(u) ∈ A. Homology groups in model theory 43 (3) (Closure under localizations) Suppose that f : X → C is in A for some X ⊆ P(s) and t ∈ X. Then f |t : X|t → C is also in A.

Representation theory of separable algebras has deeply affected development of the classical descriptive set theory, as evident from the terminology of both subjects (terms ‘smooth’ and ‘analytic’ have the same, albeit nonstandard in other areas of mathematics, meaning). Extension of the work of Glimm and Effros on representation theory combined with methods from logic initiated the abstract classification theory (§3). The representation theory of nonseparable algebras was largely abandoned because some of the central problems proved to be intractable (see the introduction to [1]).

Assume A has n-CA for some n ≥ 2. Then Hn−1 (A; B) is generated by {[c] : c is an (n − 1)-shell over B}. In particular, if any (n − 1)-shell over B is a boundary, then so is any (n − 1)-cycle. 19. If A has n-CA for some n ≥ 3, then Hn−2 (A; B) = 0. 28. 20. If n ≥ 1, an n-pocket is an n-cycle of the form f − g, where f and g are n-simplices with support S (where S is an (n + 1)-element set). 21. Suppose that f, g ∈ Sn (A) are isomorphic functors such that ∂n f = ∂n g. Then the n-pocket f − g is a boundary.