Matrix convex sets and real algebraic geometry

Convexity is a basic notion from geometry that is applied for solving problems across many sciences. In optimization, convexity leads to reliable and numerically tractable problems. Convex optimization is employed in control theory, communications and networks, signal processing, mechanical engineering, finance, optimal design in statistics, coding theory, etc. This proposal aims to determine classes of optimization problems which are effectively convex ones, even if they do not look like it. Advances in free analysis and real algebraic geometry are yielding exciting new approaches to this question, but there are still fundamental challenges ahead. This proposal intends to overcome these by using algebraic, geometric and analytic tools in a novel way. The project is designed modularly consisting of two strands, one focusing on free function theory, and the other on real algebraic geometry and positivity of noncommutative functions. We will also vigorously pursue applications of free analysis to related fields such as operator algebra and quantum information theory. Key for this will be advances in algorithms and their implementations which we intend to make available online to the wider scientific community.