Mathematical Methods in Systems Optimization and Control : Festschrift in Honor of J. William Helton Preface Inroduction (an overview of the contents of the book from the perspective of Heltons landmark contributions) Biographical Note and List of Publications of J.W. Helton Bounded Analytic Interpolation and H infinity Control Linear Matrix Inequalities and Matrix Convexity Optimization and Realization in the Free * algebra Automatic Verification of Matrix Inequalities Non commutative Algebraic Geometry Non Commutative Differential Geometry and Probability Modern Trends in Robust Control Challenges of Mathematical Biology Mathematical Programming and Computer Science. This volume is dedicated to Bill Helton on the occasion of his sixty fifth birthday. It contains biographical material a list of Bills publications a detailed survey of Bills contributions to operator theory optimization and control and 19 technical articles. Most of the technical articles are expository and should serve as useful introductions to many of the areas which Bills highly original contributions have helped to shape over the last forty odd years. These include interpolation Szegö limit theorems Nehari problems trace formulas systems and control theory convexity matrix completion problems linear matrix inequalities and optimization. The book should be useful to graduate students in mathematics and engineering as well as to faculty and individuals seeking entry level introductions and references to the indicated topics. It can also serve as a supplementary text to numerous courses in pure and applied mathematics and engineering and as a source book for seminars.