Roy Dyckhoff A cut-free sequent calculus for distributive lattices with adjoint pairs of modal operators. Abstract: We present a cut-free sequent calculus for validity of inequations m <= m' in an arbitrary distributive lattice with a family of adjoint pairs (f,g) of operators, where each f is both the left adjoint of the corresponding g and join-preserving. Such lattices are natural generalisations of Boolean algebras with (provided a symmetry axiom, i.e. B: <>[]phi implies phi, holds) the traditional operators of <> and [] from classical modal logic; we have in mind future work to quantales with such operators and applications in epistemic logic. Sequents in such a calculus are formed not from sets, multisets or lists but from more complex structures, here called ''contexts'', but variously known as ``nested sequents'' and``deep sequents''; we will try and trace some of their history. This is joint work with Mehrnoosh Sadrzadeh.