Mathematical Sciences: Combinatorial Methods in Computational Algebraic Geometry

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This award supports the research in algebraic geometry and combinatorics of Professor Bernd Sturmfels of Cornell University. Dr. Sturmfels has proposed to make use of the combinatorial structure of convex polytopes to attack problems in constructive algebra and algebraic geometry. His research on combinatorial structures arising in geometry will include a study of higher-order matroids associated with graded ideals and how they relate to the state polytopes of these ideals. 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.