Mathematical Sciences: Algebraic Shifting

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This award supports the research of Professor Helene Barcelo in the field of algebraic combinatorics. The algebraic shifting method introduced by Kalai in 1984 has proven to be an extremely successful tool for characterizing the f-vectors of a wide range of classes of simplicial complexes. The goal of this research is to investigate possible generalizations of this method to algebraic structures other than simplicial complexes. The algebraic structures to be studied share the common property that they have a class of combinatorial objects associated with them. This research is in the general area of Combinatorics. Combinatorics attempts to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of lapge networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.