J. Maurice Rojas, Texas A&M, College Station

**Complexity and Reality in Equation Solving **

We survey some recent practical and theoretical results on solving systems
of nonlinear polynomial equations over the real numbers. Such equations are
ubiquitous in fields

such as control theory, game theory, geometric modelling, optimization, mathematical
biology, and statistics, to name just a few.

The methods we focus on are based on toric varieties. Conceptually, this means that one can apply basic geometric intuition (with familiar constructs like volumes and subdivisions of polyhedra) to the underlying advanced algebraic algorithms. On a practical level, these new techniques yield exponential speed-ups over many earlier algorithms, such as those based on Grobner bases or classical resultants.

We will illustrate our toric techniques with many color pictures and animations, and we assume no background in algebraic geometry.