Mathematical Sciences: Lattice Theory and Algebraic Combinatorics

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The research involves problems in lattice theory and algebraic combinatorics, including the conjectures that finite- rank Arguesian lattices with distributive skeleton are representable, that homomorphic images of representable lattices are representable, and that the dual of a representable lattice need not be representable. Other work in lattice theory will involve the investigation of new continuous analogues of finite partition lattices. The problems in algebraic combinatorics are to prove a conjectured bijection between shifted standard Young tableau of a certain shape and maximal-length words in the Coxeter group of type B(n), and to study the applications of a new combinatorial formulation of multivariate Lagrange inversion. Lattices appear throughout mathematics, making lattice theory an important topic of study, not only for its own intrinsic interest, but for its applications in mathematics and related disciplines.