Convexity in real algebraic geometry

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In this project, problems from convexity and from convex and polynomial optimisation are studied with specific methods from real algebraic geometry. This concerns fundamental questions arising from applications in optimisation (in particular semidefinite programming and positive polynomials), but also the classical geometry of curves and surfaces and their higher dimensional analogues. Two aspects that have received a lot of attention in recent years concern the representation of real polynomials by determinants and the description of convex sets by linear matrix inequalities.