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.
