Combinatorial and Real Algebraic Geometry

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，以其理论深度和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来描述，科学和工程中概念之间的关系可以用多项式来建模。不管它们的来源是什么，一旦多项式进入人们的视野，代数几何的理论基础、经典实例的宝库和现代计算工具就可以用来解决手头的问题。该研究项目旨在通过两种方式加强代数几何作为应用工具的作用。首先，重点将放在开发代数几何和应用程序之间的接口。其次是代数几何的组合方面的工作，因为这发展了工具和思想，以更好地理解具有特殊结构的代数几何对象，这些高度结构化的对象是在代数几何和其他领域之间的相互作用中最常见的对象，无论是在数学还是在应用科学中。该项目还将包括对学生和博士后的培训，研究者通过数学圈在当地开展外联活动，以及他与尼日利亚数学的持续互动。通常，在数学或科学的其他部分中出现的代数几何对象具有很强的组合结构（例如，环变或格拉斯曼），或者应用程序需要真正的解决方案。因此，本项目在组合几何和实代数几何领域的研究将有助于促进我们对这些主题的基本理解，并为应用奠定基础。该项目涉及代数几何中的三个主题：实环变分、热带几何和舒伯特微积分。在这些领域中，研究者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，再到从应用程序的问题中获得灵感。特别值得注意的是，我们的目标是发展一个丰富而稳健的不合理环变理论，理解热带物体补的几何和拓扑，通过一个有用的位移对偶等价理论在C型舒伯特微积分中建立正性，以及理解舒伯特微积分中的伽罗瓦群。该奖项由代数、数论和组合学项目共同资助。

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.