Real algebraic and combinatorial structures in matrix spaces

该项目旨在应用数学工具来研究优化、工程和统计学方面的对象。所涉及的研究将导致对基础数学的更好理解，并进一步开发和分析算法。这种理论的发展对数学界和更广泛的科学界都有好处，并加强了它们之间的联系。这个项目的另一个重要部分是指导本科生和研究生。参与该项目将使学生接触到广泛的数学领域，使他们深刻认识到跨学科研究的重要性，并培养他们开展跨学科研究所需的理论和计算技能。所提出的研究的广泛目标是发展实际代数几何和组合学中的计算技术，以便研究优化和其他应用中出现的对象。真正的代数几何是研究多项式方程和不等式的解，它可以描述工程和科学领域的大量系统。热带几何是一个强大的工具，用于从实际代数集中提取离散数据，以及构造具有期望性质的实际代数集的例子。本研究将这些强大的数学工具应用于线性代数和凸优化问题。稳定多项式、行列式和矩阵的线性空间是这些领域的基本对象。更好地理解它们的基本几何结构将推进这些领域以及它们之间的联系。

This project aims to apply mathematical tools to study objects from optimization, engineering, and statistics. The research involved will lead to a better understanding of the underlying mathematics and further the development and analysis of algorithms. Development of such theory benefits both the mathematical and broader scientific communities and strengthens the connections between them. Another important part of this project involves mentoring undergraduate and graduate students. Participation in this project will expose students to a wide range of fields in mathematics, impress upon them the importance of interdisciplinary research, and train them in the theory and computational skills needed to carry it out.The broad goals of the proposed research are to develop computational techniques in real algebraic geometry and combinatorics in order to study objects arising in optimization and other applications. Real algebraic geometry is the study of solutions to polynomial equations and inequalities, which can describe a vast array of systems from engineering and science. Tropical geometry is a powerful tool for extracting discrete data from real algebraic sets as well as for constructing examples of real algebraic sets with desired properties. The proposed research applies these powerful mathematical tools to problems in linear algebra and convex optimization. Stable polynomials, determinants, and linear spaces of matrices are fundamental objects in these fields. A better understanding of their underlying geometry would advance these fields and the connections between them.