Combinatorial and Real Algebraic Geometry

组合和实代数几何

• 批准号: 1501370
• 负责人: Frank Sottile
• 金额: $ 34.76万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501370&HistoricalAwards=false
• 关键词: Combinatorial; Real; Algebraic; Geometry

Algebraic geometry, which is the mathematical study of solutions to systems of polynomial equations, is a core mathematical discipline noted for its theoretical depth and its interactions with other areas of mathematics. Algebraic geometry is also a tool for applications, since physical objects can be described by polynomial equations, and relations between concepts in science and engineering may be modeled by polynomials. Whatever their source, once polynomials enter the picture, the theoretical base, trove of classical examples, and modern computational tools of algebraic geometry may be brought to bear on the problem at hand. This research project aims to strengthen the role of algebraic geometry as a tool for applications, in two ways. First, a major focus will be on developing the interface between algebraic geometry and applications. Second is work on combinatorial aspects of algebraic geometry, for this develops tools and ideas to better understand objects in algebraic geometry with special structure, and these highly structured objects are those that appear most commonly in the interactions between algebraic geometry and other fields, both within mathematics and in the applied sciences. This project will also involve training of students and postdocs, the investigator's outreach activities at the local level through a mathematics circle, and his continued interaction with mathematics in Nigeria. Oftentimes, objects from algebraic geometry that arise in other parts of mathematics or science have strong combinatorial structures (e.g., toric varieties or Grassmannians) or else the application demands real solutions. Consequently, the research in areas of combinatorial and real algebraic geometry in this project will serve both to advance our basic understanding of these topics and to help build a foundation for applications. This project involves research in three topics within algebraic geometry: real toric varieties, tropical geometry, and Schubert calculus. In each of these areas the investigator will work with collaborators and students on projects ranging from developing foundations to understanding key examples to work inspired by problems from applications. Of particular note are the goals of developing a rich and robust theory of irrational toric varieties, understanding the geometry and topology of complements of tropical objects, establishing positivity in type C Schubert calculus via a useful theory of shifted dual equivalence, and understanding Galois groups in the Schubert calculus. This award is jointly funded by the Algebra and Number Theory and Combinatorics programs.

代数几何是对多项式方程组解的数学研究，是一门核心数学学科，以其理论深度及其与其他数学领域的相互作用而闻名。代数几何也是一种应用工具，因为物理对象可以用多项式方程描述，科学和工程中概念之间的关系可以用多项式建模。无论它们的来源是什么，一旦多项式进入画面，理论基础，经典例子的宝藏，以及代数几何的现代计算工具可能会对手头的问题产生影响。这个研究项目的目的是加强作用的代数几何作为一种工具的应用，在两个方面。首先，一个主要的重点将是发展代数几何和应用之间的接口。其次是工作的组合方面的代数几何，为这发展的工具和想法，以更好地了解对象的代数几何与特殊结构，这些高度结构化的对象是那些最常见的互动之间的代数几何和其他领域，无论是在数学和应用科学。该项目还将涉及对学生和博士后的培训，调查员通过数学圈在地方一级的推广活动，以及他与尼日利亚数学界的持续互动。通常，在数学或科学的其他部分中出现的来自代数几何的对象具有强组合结构（例如，复曲面品种或Grassmannians），否则应用需要真实的解决方案。因此，在组合和真实的代数几何领域的研究在这个项目中将有助于既推进我们对这些主题的基本理解，并帮助建立应用的基础。该项目涉及代数几何中的三个主题的研究：真实的复曲面品种，热带几何和舒伯特演算。在每个领域，研究人员将与合作者和学生一起开展项目，从开发基础到理解关键示例，再到受应用问题启发的工作。特别值得注意的是发展一个丰富和强大的理论的非理性环面品种的目标，了解几何和拓扑结构的补充热带物体，建立积极性在C型舒伯特演算通过一个有用的理论转移对偶等价，并了解伽罗瓦集团在舒伯特演算。该奖项是由代数和数论和组合学计划共同资助的。