Bernd Sturmfels: (UC Berkeley, Math)
In this lecture we discuss a general theorem which states that a system of algebraic equations and inequalities either has a solution point or there exists a certain certificate which obviously shows that no solution exists. In the special case when "algebraic" is replaced by "linear", this statement is the familiar Duality Theorem of Linear Programming. Recent work of Parrilo and others establishes a fascinating connection to Semidefinite Programming.