Geometry of Objects with Many Vertices

Magnus Goffeng received his Ph.D. in Mathematics at the University of Gothenburg in 2011. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position in Professor Ryszard Nest’s research group at the Copenhagen University, Denmark.

The word geometry brings to mind triangles and circles. Mathematicians think of more abstract objects in higher dimensions as well. The common feature of the objects of study here is that they are not smooth but edgy. Examples of smooth curves are circles, while triangles are examples of curves with vertices.

Vertices cause difficulties when one tries to use analytical methods. In order to apply such techniques, mathematicians transform objects with vertices into smoother ones by rounding out sharp points. Sometimes such a smoothing procedure is not possible though, there is no general guarantee that it can be done. An example of a curve too thorny to be smoothed out was given by Karl Weierstrass in 1872. Surprisingly, the graph of Weierstrass function consists only of vertices.

The goal of Magnus Goffeng’s research project is to develop robust analytical methods to study of geometric objects which are not smooth, by using techniques from the branch of mathematics called non-commutative geometry. This was developed in the 1970’s by the French mathematician and Fields Medal recipient Alain Connes. The inspiration for such techniques comes from quantum mechanics and from quantum field theory. Also tools from many branches of mathematics including topology, differential geometry and analysis, are necessary for the project.

Photo: Petra Lein