PECASE: Combinatorial Structures in Algebra and Geometry

翻译后摘要：这个职业奖支持舒伯特品种的组合结构的研究，主要是通过研究他们的奇点。舒伯特品种是迷人的几何对象在于交叉的几个领域的数学;他们的研究开始与经典作品中的射影几何的十九世纪。 更好地理解这些结构将对代数几何、组合数学和表示论产生影响。在数学之外，这些结果可能在理论物理学、计算机图形学和巴基球（碳-60分子）的研究中有应用。 这个提议的另一个目标是探索计算机在数学中的变化。 计算机验证和计算机证明在通过模式回避来表征舒伯特簇的性质方面起着核心作用。数学教育和研究有着内在的联系-研究带来了学习新数学概念所需的创造力和动力。 特别是，本科生喜欢面对未解决的问题的挑战。 然而，由于数学研究的性质，找到“好”的本科研究问题是困难的。 计算机验证证明和计算机实验特别适合本科生的技能和知识，同时也对研究生和更高级的研究人员感兴趣。 因此，这一研究方向已通过新课程和本科生研究项目纳入提案的教育部分。

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.