# Research-Initiatives

## USAG Uniformized structures in arithmetic and geometry

How can complicated geometric spaces be described by simpler spaces? An approach is offered by the concept of uniformization, which has its origin in the works of the famous 19th century mathematicians Bernhard Riemann and Felix Klein. This approach offers the possibility of substituting a complicated geometric space by a much simpler one without changing the local structure. The complexity is described by the inner symmetries of the simpler space. This basic idea has proved to be exceedingly powerful and has been generalized in various directions, e.g. in so-called non-Archimedean geometries. New insights are expected to be gained regarding current arithmetic and geometric problems of classification in the LOEWE research cluster “Uniformized Structures in Arithmetic and Geometry” by connecting various techniques of uniformization. The investigations in this undertaking will focus on algebraic varieties, i.e. solution sets of equation systems given by polynomials. Important examples such as elliptical curves and Calabi-Yau varieties also play a prominent role in applications in the areas of cryptography and mathematical physics. Cryptographic algorithms, which are based on elliptical curves and related number-theoretic structures, are used, for example, in encryptions employed for handling online shopping and mobile phone calls.

### Partners

- Technical University Darmstadt
- Goethe-University Frankfurt am Main

### Fields of study

- Working Group Algebra, FB Mathematics, TU Darmstadt;

- Focus on Algebra and Geometry, FB Computer Science and Mathematics, GU Frankfurt

### Funding period

since 2018### Project Coordinator

- Prof. Dr. Jan Hendrik Bruinier,

Technical University Darmstadt

### Locations

- Darmstadt
- Frankfurt am Main