Grigoriy Blekherman is an assistant professor at Georgia Institute of Technology and a 2012 recipient of the Sloan Research Fellowship. His research interests lie at the intersection of convex and algebraic geometry. Pablo A. Parrilo is a Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. He has received the SIAG/CST Prize and the IEEE Antonio Ruberti Young Researcher Prize. His research interests include mathematical optimization, systems and control theory, and computational methods for engineering applications. Rekha R. Thomas is a Professor of Mathematics at the University of Washington. Her research interests are in optimization and computational algebra.

• List of NotationChapter 1: What is Convex Algebraic Geometry?Chapter 2: Semidefinite OptimizationChapter 3: Polynomial Optimization, Sums of Squares, and ApplicationsChapter 4: Nonnegative Polynomials and Sums of SquaresChapter 5: DualitiesChapter 6: Semidefinite RepresentabilityChapter 7: Convex Hulls of Algebraic SetsChapter 8: Free ConvexityChapter 9: Sums of Hermitian Squares: Old and NewAppendix A: Background Material