Unknown links between geometric objects

Dimitri Leemans is a professor at the Université Libre de Bruxelles, Belgium. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics and Mathematical Statistics, Umeå University.

Ever since the work of German mathematician Felix Klein in the nineteenth century, mathematicians have studied the geometric properties of an object by analyzing their symmetries. A symmetry is a transformation of an object under which it keeps its shape. For example, an object is mirror symmetric if it is identical to its mirror image, it is rotationally if it can be rotated around a given axis without changing shape. A square is an example of an object that demonstrates both mirror symmetry and rotational symmetry. Together, the symmetries of an object form an algebraic symmetry group. 

There are many different geometric spaces in mathematics, each of which has its own algebraic symmetry group. The links between them have not yet been fully understood, and the subject of this project is the discovery of links between geometric objects that initially appear to be entirely different. 

In Belgium, about 50 years ago, Jacques Tits started his studies – in the spirit of Klein – to link algebraic symmetry groups to symmetries in geometric objects. Tits succeeded in classifying a specific group of symmetries through the effect they had on geometric objects in the same space. The aim of the project is to generalize Tits’ theory and find new geometric objects with more, and more exotic, symmetries.