PECASE: Combinatorial Structures in Algebra and Geometry

摘要：该奖项主要通过研究舒伯特品种的奇异性来支持对其组合结构的研究。舒伯特变分是令人着迷的几何对象，它位于几个数学领域的交叉点；他们的研究始于19世纪射影几何的经典作品。更好地理解这些结构将对代数几何、组合学和表示理论产生影响。在数学之外，这些结果可以应用于理论物理、计算机图形学和巴基球（碳-60分子）的研究。本提案的进一步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表征舒伯特品种的特性方面起着核心作用。这项建议的一个重要方面是其教育内容。数学教育和数学研究有着内在的联系，研究可以激发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问题的挑战。然而，由于数学研究的性质，找到“好”的本科生研究问题是困难的。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，这条研究路线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.