Algebraic Geometry: Computations and Applications

线性代数及其数值方法是科学计算的基础，并为应用数学的广泛研究提供了信息。代数几何是非线性代数的几何；它的主要对象是由多个变量的多项式的集合描述的集合。这类集合包括变种和半代数集合，在许多情况下都会出现，特别是在最优化、统计学、金融学和数量生物学中。最近在理论和计算资源方面的进展使研究人员能够将一些可用于线性模型的技术扩展到非线性模型。这个项目开发了基于代数几何的新方法，用于符号和数值计算，以解决此类应用中出现的问题。介于线性和非线性代数之间的中间步骤包括分段线性代数(特别是极大代数和热带几何)和多线性代数(特别是张量及其分解的研究)。这个项目建立在这些联系的基础上。它的四个主要主题是极大似然几何、欧几里德距离优化、凸代数几何和经典模空间。具体目标包括确定行列式变元的最大似然程度，有界非负秩矩阵的半代数刻画，欧氏距离度问题的松弛平方和的发展，以及Gram谱面体的几何刻画。模空间的新算法的设计和实现，例如Del Pezzo曲面，形成了通往核心代数几何研究社区的桥梁。

Linear algebra and its numerical methods are the foundation of scientific computing and inform a wide range of research in applied mathematics. Algebraic geometry is the geometry of non-linear algebra; its primary objects are sets described by collections of polynomials in several variables. Such sets, which include varieties and semialgebraic sets, arise in many contexts, notably in optimization, statistics, finance, and quantitative biology. Recent advances, both in theory and in computing resources, have enabled researchers to extend some of the techniques available for linear models to non-linear models. This project develops new methodologies based on algebraic geometry for both symbolic and numerical computing to tackle problems arising in such applications. Intermediate steps between linear and non-linear algebra include piecewise-linear algebra (in particular, the max-plus algebra and tropical geometry) and multilinear algebra (in particular, the study of tensors and their decompositions). This project builds on these connections. Its four main themes are maximum likelihood geometry, Euclidean distance optimization, convex algebraic geometry, and classical moduli spaces. Concrete goals include the determination of the maximum likelihood degrees of determinantal varieties, the semialgebraic characterization of matrices with bounded nonnegative rank, the development of sum of squares relaxations for the Euclidean distance degree problem, and a geometric characterization of Gram spectrahedra. The design and implementation of novel algorithms for moduli spaces, for example for del Pezzo surfaces, forms a bridge to the research communities in core algebraic geometry.