Convex algebraic geometry, optimization and applications

September 21 to September 25, 2009

at the

American Institute of Mathematics, San Jose, California

organized by

William Helton and Jiawang Nie

Original Announcement

This workshop 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.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

The moment problem for continuous positive semidefinite linear functionals

by Mehdi Ghasemi, Salma Kuhlmann, and Ebrahim Samei, Arch. Math. (Basel) 100 (2013), no. 1, 43-53 MR3009662

Quartic curves and their bitangents

by Daniel Plaumann, Bernd Sturmfels, and Cynthia Vinzant

Dualities in convex algebraic geometry

by Philipp Rostalski and Bernd Sturmfels

Positivity and optimization for semi-algebraic functions

by Jean-Bernard Lasserre and Mihai Putinar

Convex Forms that are not Sums of Squares

by Grigoriy Blekherman