PECASE: Combinatorial Structures in Algebra and Geometry

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Abstract:This CAREER award supports research on the combinatorial structures of Schubert varieties, primarily by studying their singularities. Schubert varieties are fascinating geometrical objects which lie at the intersection of several fields of mathematics; their study began with the classical works in projective geometry of the nineteen century. Better understanding of these structures would have impact in algebraic geometry, combinatorics and representation theory. Outside of mathematics, these results may have applications in theoretical physics, computer graphics, and the study of Bucky balls (Carbon-60 molecules). A further goal of this proposal is to explore the changing roll of computers in mathematics. Computer verification and computer proofs play a central role in characterizing properties of Schubert varieties by pattern avoidance.An important aspect of this proposal is its educational component. Mathematics education and research are intrinsically linked - research brings out the creativity and drive needed to learn new mathematical concepts. In particular, undergraduate students enjoy the challenge of facing unsolved problems. However, due to the nature of mathematics research, finding "good" undergraduate research problems is difficult. Computer verified proofs and computer experimentation are particularly well suited to the skills and knowledge of undergraduates while also of interest to graduate students and more senior researchers. Therefore, this line of research has been incorporated into the education portion of the proposal through a new course and undergraduate research projects.