Greg Reid, Department of Applied Mathematics, Western University

Numerical Differential-Geometric Algorithms in Computational Discovery
We consider classes of models specified by polynomial equations, or more generally polynomially nonlinear partial differential equations, with parametric coefficients. A basic task in computational discovery is to identify exceptional members of the class characterized by special properties, such as large solution spaces, symmetry groups or other properties. Symbolic algorithms such as Groebner Bases and their differential generalizations can some times be applied to such problems. These can be effective for systems with exact (e.g. rational) coefficients, but even then can be prohibitively expensive. They are unstable when applied to approximate systems. I will describe progress in the approximate case, in the new area of numerical algebraic geometry, together with fascinating recent progress in convex geometry, and semi-definite programming methods which extends such methods to the reals. This is joint work with Fei Wang, Henry Wolkowicz and Wenyuan Wu.