Convexity in an Abstract Setting

Professor Bo Berndtsson will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Complex Analysis in Several Variables Group at the Department of Mathematical Science, Chalmers University of Technology and the University of Gothenburg.

In common language, a solid is convex if it bulges out or when it has no concavities. At the same time convexity is such an important concept that it is the subject of research in an own area of mathematics called convex geometry. It has important applications in other branches of mathematics as well.

An important cornerstone of convex geometry was laid over a hundred years ago with the establishment of Brunn-Minkowski theorem called also Brunn-Minkowski inequality. It has spawned research and a multitude of problems studied ever since.

The proposed project centers on a theorem which generalizes the Brunn-Minkowski theorem to spaces of functions over complex numbers as opposed to volumes in spaces over real numbers in the original theorem. The generalization has proven very fruitful in the study of complex geometry, i.e. geometry of complex manifolds. Such manifolds have important applications in modern physics as well as in other branches of mathematics including algebraic geometry and number theory.

The project will focus both on further extensions of the theorem and on its new applications, which have already shown important implications for algebraic geometry. Also the relations between convexity and complex geometry will be further studied. This will lead to further expansion into new aspects of Brunn-Minkowski inequality.