Mathematical Sciences: Algebraic Combinatorics

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This award supports the research in Combinatorics of Professor Philip Hanlon of the University of Michigan at Ann Arbor. Dr. Hanlon will work in two general problem areas at the junction of Combinatorics and Representation Theory. The first area is concerned with the homology of certain nilpotent Lie algebras, and homological extensions to the Macdonald root- system conjectures. The second area is concerned with problems related to Jones's basic construction, and the algebra structures of Brauer's centralizer algebras corresponding to Stanley's Fibonacci r-differential posets. This research falls in the broad category of Combinatorics, which is one of the most active fields in today's mathematics. Fundamentally, Combinatorics represents a systematization of the very first of all mathematical activities, counting. In its modern development, however, Combinatorics has gone beyond just counting to make use of a wide variety of advanced mathematical techniques, and although its roots go back several centuries, the field has had an explosive development in the past few decades because of its importance in communications and information technology.