A great challenge in complex geometry

Professor David Witt Nyström will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg.

A vital question in geometry is understanding when a geometric object can be given an optimal shape. It was not until 1907 that Paul Koebe and Henri Poincaré proved almost simultaneously that two-dimensional surfaces can always be deformed into a shape with constant curvature.

The development of the proof of this uniformisation theorem had occured throughout the nineteenth century in parallel with the emergence of modern algebraic geometry, the birth of complex analysis and topology. Even more new tools were necessary to prove in 2003 a three-dimensional equivalent of the uniformisation theorem – Thurston’s geometrisation conjecture.

Also in complex geometry, the subject of this project, the question of when objects can be given an optimal shape is raised. Here, the geometric objects are defined using complex numbers and the issue of uniformisation, known as the Yau-Tian-Donaldson conjecture, is one of the major unsolved problems in complex geometry.

A promising way to approach the solution is to use methods from non-Archimedean geometry, where the familiar real and complex numbers are replaced by abstract objects called non-Archimedean fields. These methods have proved successful in one important
special case, but much remains to be done, including showing that the non-Archimedean Monge-Ampère equation is solvable.

The aim of the project is to prove that the equation is solvable and thus get one step closer to proving the Yau-Tian-Donaldson conjecture.