Logic, stability and perturbation theory: novel bridges between the foundations of mathematics and operator algebras

As all other sciences rely on mathematics, I think of science as a building, with the ground floor being made up by mathematics. As we want to keep the building in good shape so it can grow in a creative, new and strong way, we need to take care of the foundations. Logic represents these foundations. It provides the framework for mathematics (consisting of the assumptions we work with) and general abstract tools for considering mathematical structures and languages. The stronger and deeper we make the connections between these foundations and the building, the healthier the building will grow. Logic has had a strong impact across mathematics, particularly in algebra and geometry. In this project, I aim to establish new links between logic and the area of mathematical analysis --- the study of the infinite --- where to date the connections are less well developed. There is an exciting opportunity to bring very new techniques to bear on long standing questions.