Auslander–Reiten conjecture
Do vanishing higher extension groups force a module to be projective?
Algebras and modules
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Finitistic dimension conjecture
For each finite-dimensional algebra, are finite projective dimensions of finitely generated modules bounded?
Amenability of Thompson’s group F
Can Thompson’s group F admit an average that is unchanged by every group translation?
UCT problem for nuclear C*-algebras
Does every separable nuclear C*-algebra satisfy the universal coefficient theorem relating its K-theoretic invariants?
Alperin weight conjecture
Can each block’s irreducible representations in prime characteristic be counted entirely from local subgroup data?
Broué’s abelian defect group conjecture
Does an abelian-defect representation block retain its derived structure when replaced by its local normalizer block?