Convex algebraic geometry, optimization and applications

September 21 to September 25, 2009

at the

American Institute of Mathematics, Palo Alto, California

organized by

William Helton, Jiawang Nie, and Pablo Parrilo

This workshop, sponsored by AIM and the NSF, will be devoted to the study of ``Convex Algebraic Geometry'' and some of its numerous applications. Convexity plays a fundamental role in mathematics, and its ubiquity in optimization makes it of crucial importance in many domains of application. In such situations, the geometric properties of convex sets are complemented by additional algebraic structure (e.g., the semialgebraic case, where sets are defined by means of polynomial inequalities). In this case, the rich interactions between the geometric, algebraic, and computational aspects are not yet well-understood.

Falling into this setting are classical linear programming (LP), the more recent area of semidefinite programming (SDP), and the associated linear matrix inequalities (LMI),which have had a major impact on engineering systems, combinatorial optimization and other areas. One focus of the workshop is the study (arising from linear systems engineering) of polynomials in matrices whose form does not depend on the size of the matrices; this requires development of a noncommutative semialgebraic geometry.

Convex Algebraic Geometry involves a healthy combination of real algebraic geometry, functional analysis, operator theory, convex optimization and several areas of application. We expect a diverse group with common emerging interests.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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