PECASE: Combinatorial Structures in Algebra and Geometry

PECASE：代数和几何中的组合结构

• 批准号: 0437359
• 负责人: Sara Billey
• 金额: --
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-12-01 至 2006-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0437359&HistoricalAwards=false
• 关键词: PECASE; Combinatorial; Structures; Algebra; Geometry

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.

摘要：该职业奖主要通过研究舒伯特簇的奇点来支持对其组合结构的研究。舒伯特变种是一种迷人的几何对象，它位于几个数学领域的交叉点上；它们的研究始于19世纪射影几何的经典著作。更好地理解这些结构将对代数几何、组合学和表示论产生影响。在数学之外，这些结果可能会在理论物理、计算机图形学和巴基球(碳-60分子)的研究中得到应用。这项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避免来刻画Schubert变种的性质方面起着核心作用。这一提议的一个重要方面是它的教育成分。数学教育和研究本质上是联系在一起的--研究激发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，找到好的本科生研究问题是困难的。计算机验证的证明和计算机实验特别适合本科生的技能和知识，同时也是研究生和更资深的研究人员感兴趣的。因此，这一研究路线已经通过新课程和本科研究项目被纳入提案的教育部分。