Mathematical Sciences: Lattice Theory and Algebraic Combinatorics

该研究涉及格论和代数组合中的问题，包括具有分布骨架的有限秩格是可表征的，可表征格的同态象是可表征的，以及可表征格的对偶不一定是可表征的。格理论的其他工作将涉及对有限分格的新的连续类似物的研究。代数组合学中的问题是证明一定形状的移位标准Young表与B(n)型Coxeter群中最大长度词之间的猜想双射，以及研究多元拉格朗日反演的一种新的组合公式的应用。格在数学中无处不在，使格理论成为一个重要的研究课题，不仅因为它本身的内在兴趣，而且因为它在数学和相关学科中的应用。

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.