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.

代数几何形状是对多项式方程系统解决方案的数学研究，是一门核心数学学科，其理论深度及其与其他数学领域的相互作用。代数几何形状也是应用程序的工具，因为物理对象可以通过多项式方程来描述，科学和工程学中的概念之间的关系可以由多项式建模。无论它们的来源如何，一旦多项式进入图片，理论基础，经典示例的trove以及代数几何的现代计算工具就可以带来手头的问题。该研究项目旨在通过两种方式加强代数几何形状作为应用的工具的作用。首先，重点是开发代数几何和应用之间的接口。其次是代数几何形状的组合方面的工作，因为这开发了具有特殊结构的代数几何形状中的对象，而这些高度结构化的对象是在代数几何形状和数学和应用科学中最常见的相互作用中最常见的对象。该项目还将涉及对学生和博士后的培训，通过数学圈在当地的调查人员的外展活动以及他与尼日利亚数学的持续互动。通常，在数学或科学其他部分中出现的代数几何形状的对象具有强大的组合结构（例如，福利品种或司羊植物），或者应用程序需要真实的解决方案。因此，该项目中对组合和实际代数几何形状领域的研究将既可以提高我们对这些主题的基本理解，又有助于为应用建立基础。该项目涉及代数几何形状内三个主题的研究：真实的曲折品种，热带几何形状和舒伯特·微积分。在这些领域的每个领域中，研究者将与合作者和学生合作进行项目，从发展基础到了解关键示例到受应用程序问题的启发的工作。特别值得注意的是，建立一个不合理的感谢您品种的丰富而坚固的理论，了解热带物体补充物的几何形状和拓扑，并通过转移的双重等效性的有用理论在C型C舒伯特型中建立阳性，并了解Schubert Calculus中的Galois群体。该奖项由代数理论和组合计划共同资助。