Recent advances in both the theory and implementation of computational algebraic geometry have led to new, striking applications to a variety of fields of research. The articles in this volume highlight a range of these applications and provide introductory material for topics covered in the IMA workshops on "Optimization and Control" and "Applications in Biology, Dynamics, and Statistics" held during the IMA year on Applications of Algebraic Geometry. The articles related to optimization and control focus on burgeoning use of semidefinite programming and moment matrix techniques in computational real algebraic geometry. The new direction towards a systematic study of non-commutative real algebraic geometry is well represented in the volume. Other articles provide an overview of the way computational algebra is useful for analysis of contingency tables, reconstruction of phylogenetic trees, and in systems biology. The contributions collected in this volume are accessible to non-experts, self-contained and informative; they quickly move towards cutting edge research in these areas, and provide a wealth of open problems for future research.

Polynomial Optimization on Odd-Dimensional Spheres.- Engineering Systems and Free Semi-Algebraic Geometry.- Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation.- Using Invariants for Phylogenetic Tree Construction.- On the Algebraic Geometry of Polynomial Dynamical Systems.- A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals.- Sums of Squares, Moment Matrices and Optimization Over Polynomials.- Positivity and Sums of Squares: A Guide to Recent Results.- Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas.- Open Problems in Algebraic Statistics.