Real Algebraic Geometry and its Applications

TL;DR

This survey explores real algebraic geometry, its foundational concepts, and recent applications in optimization, especially semidefinite programming, highlighting geometric problems and emerging non-commutative theories.

Contribution

It provides a comprehensive overview of real algebraic geometry and connects classical theory to modern optimization and non-commutative geometry developments.

Findings

Connections between real algebraic geometry and semidefinite programming

Classification of feasible sets in semidefinite programming

Emerging research in non-commutative real algebra and geometry

Abstract

This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to optimization, mostly via semidefinite programming. We introduce interesting geometric problems arising from the classification of feasible sets for semidefinite programming. We close with a perspective on the very active area of non-commutative real algebra and geometry.