Applied and Computational Algebraic Geometry

This PhD proposal centers on systems of polynomial equations that mathematically model several problems in network reliability theory, rigidity theory, and statistics. The main idea is to exploit combinatorial and geometric structures in these systems and use them to efficiently study their solutions spaces. Solving systems of polynomials in general is extremely difficult, however, in the aforementioned applications, the main problem is reduced to certifying the existence of a positive or real solution. In particular, such systems have a canonical underlying graph that captures the geometric information of polynomials. We will restrict our attention to such families of equations to develop new tools in real algebraic geometry. We will exploit the combinatorial structures of these graphs and use them to find efficient algorithms. In particular, we will use problem-specific insights along with tools in numerical and computational algebraic geometry to decompose the solution spaces (varieties) into more manageable components, hence speeding up the solution procedure.