The LOEWE focus group on “Uniform Structures in Arithmetic and Geometry” addresses the fundamental question of whether complex geometric spaces can be described with simple spaces in order to open up new fields of application, for example in mathematical physics. The focus group includes mathematics teams from the Technische Universität Darmstadt and the Goethe University Frankfurt.
The concept of uniformisation makes it possible to replace a complicated geometric space with a much simpler one without changing the local structure. The complexity is then described with inner symmetries of the simpler space. This basic idea has proven to be very effective. The aim of the LOEWE focus group is to gain new insights into current arithmetic and geometric classification problems by combining different techniques of uniformisation.
The investigation focuses on algebraic varieties, i.e. sets of solutions of a system of polynomial equations. Important examples, such as elliptic curves, also play an important role in applications in cryptography and mathematical physics.