Applicable Algebraic Geometry: Real Solutions, Applications, and Combinatorics

该项目的主要目标是进行实代数几何和舒伯特微积分的基础研究，培养下一代数学科学家，深化数学与应用科学之间的联系。它将支持Sottile''‘’在实代数几何和舒伯特微积分方面的基础研究以及他在代数几何应用方面的跨学科工作。这些都有一个实质性的组合组件和一个基本的计算/实验核心。这项活动将通过实验和猜想，获得更严格的多项式上界，并通过发展结构化多项式系统的实数解的下界理论，从而更好地理解多项式方程系统的实数解。它还将导致代数几何的思想和技术向其他领域的传播，并从这些领域的问题中获得启发，发展新的数学方向。其中一些工作将由Sottile''‘’在德克萨斯农工大学的研究小组进行，该小组由学生和博士后组成，并将包括大量的实验。这种研究和训练的团队方法有意识地模仿了其他科学的工作模式。此外，还将支援Sottile''‘’组织大型国际和跨学科科学会议的工作。虽然代数几何关注的是关于方程解的基本问题，但它对其他学科的价值是通过具体对象和计算工具实现的，因为应用需要特定几何对象和显式（通常是实数）解的知识。计算代数几何的现代工具具有巨大的应用潜力，但它们的使用需要协调一致的努力，才能将这项技术转移到应用科学家手中。这个项目旨在促进这种技术转让。

The main goals of this project are basic research in real algebraic geometry and the Schubert calculus, training the next generation of mathematical scientists, and deepening ties between mathematics and the applied sciences. It will support Sottile''''s basic research in real algebraic geometry and Schubert calculus and his interdisciplinary work on applications of algebraic geometry. These all have a substantial combinatorial component and an essential omputational/experimental core. This activity will result in a better understanding of real solutions to systems of polynomial equations through experimentation and conjectures by obtaining tighter fewnomial upper bounds and by developing a theory of lower bounds for the number of real solutions to structured polynomial systems. It will also result in the dissemination of ideas and techniques from algebraic geometry to other fields and the development of new directions in mathematics inspired by problems from these fields. Some of this work will be carried out by Sottile''''s research team at Texas A&M consisting of students and postdocs, and will include substantial experimentation. This team approach to research and training is consciously modeled on work patterns in the other sciences. It will also support Sottile''''s efforts organizing large-scale international and interdisciplinary scientific meetings.While algebraic geometry is concerned with basic questions about solutions to equations, its value to other disciplines is through concrete objects and computational tools, as applications require knowledge of specific geometric objects and explicit, often real-number, solutions. Modern tools from computational algebraic geometry have great potential in applications, but their use requires a concerted effort to transfer this technology into the hands of applied scientists. This project is intended to facilitate this technology transfer.