There has been notable success in applying the tools of algebraic geometry to a selection of approximation and optimisation problems. In optimisation, a whole new field of convex algebraic geometry has emerged based on the ideas of semidefinite programming relaxations of polynomial problems pioneered by J.-B. Lasserre. However, some notoriously difficult problems are still open, for instance, the optimisation problems arising from multivariate polynomial approximation, and Smale’s 7th problem, need new approaches that combine approximation, optimisation and algebraic geometry. The work in numerical analysis and approximation has been motivated by the success of low rank matrix approximations based on the matrix singular value decomposition which is related to principal component analysis. Yet, these perspective have seldom been combined into a unified approach, but experts have largely been working in parallel. This workshop aims at bringing together these alternative perspective from all three areas to develop collaborative investigations of these problems.

Algebraic Geometry, Approximation and Optimisation program at the Creswick campus of the University of Melbourne took place 5-16 February 2018 and was co-organised by the Centre for Informatics and Applied Optimisation, RMIT University and Swinburne University of Technology. It included a series of lectures by world leading experts Enrico Carlini (Politecnico di Torino), Anand Rajendra Deopurkar and Markus Hegland (Australian National University), Wolfgang Hackbusch (Max Planck Institute), Ludmila Polyakova (Saint-Petersburg State University) and other intensive group research sessions. A number of CIAO researchers were involved in these sessions which have resulted in extended international research collaboration. There has been very positive feedback from the participants to both the research program and for the facilities at Creswick.