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：该职业奖支持有关舒伯特品种组合结构的研究，主要是通过研究其奇异性。舒伯特的品种令人着迷的几何物体，位于数学几个领域的交集。他们的研究始于十九世纪的投射几何形状的古典作品。 对这些结构的更好理解将在代数几何，组合和代表理论中产生影响。在数学之外，这些结果可能在理论物理，计算机图形和Bucky Balls（碳-60分子）的研究中有应用。 该建议的另一个目标是探索数学中计算机卷的变化。 计算机验证和计算机证明在表征舒伯特品种的属性方面起着核心作用。该提案的重要方面是其教育组成部分。数学教育和研究本质上是联系的 - 研究带来了学习新数学概念所需的创造力和动力。 特别是，本科生享受面临未解决问题的挑战。 但是，由于数学研究的性质，很难找到“良好”的本科研究问题。 经过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，同时对研究生和更多的高级研究人员感兴趣。 因此，通过新课程和本科研究项目将这一研究纳入了提案的教育部分。