Applications of Algebraic Geometry

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Research in algebraic geometry is concerned with the solution sets to systems of algebraic equations, and it offers tools and models for non-linear problems across the mathematical spectrum. It now has numerous applications in the sciences and in engineering. This project is aimed at four specific directions: Spectrahedra and Orbitopes, Algebraic Statistics, Tropical Geometry, and Implicitization of Higher-Dimensional Varieties. Spectrahedra and Orbitopes are fundamental objects in convex algebraic geometry, an emerging interdisciplinary direction that focuses on convex bodies with a distinguished algebraic representation. It is driven by applications in convex optimization, especially in the theory of semidefinite programming. The project will cement the development of convex algebraic geometry, and it will lead to new semi-numerical algorithms for non-linear optimization problems with a special algebraic structure. Such problems arise frequently in Algebraic Statistics. That area is concerned with statistical models that can be represented by polynomials in the model parameters. These include graphical models for both Gaussian and discrete random variables. The project will resolve fundamental questions concerning these models, and it promises novel computational tools for statistical inference. Tropical Geometry is a piecewise-linear version of algebraic geometry that is custom-taylored for modeling applied problems. The project is aimed at key open problems on the combinatorial side of the field. Implicitization in Higher Dimensions is a new paradigm for computer algebra, and it will lead to new tools for finding the defining equations of parametrically specified algebraic varieties.