# 3rd funding period for SFB Transregio 109

## New methods for architecture, data analysis and computer graphics

Success for the mathematical Collaborative Research Center "Discretization in Geometry and Dynamics" (SFB/TRR 109): The Deutsche Forschungsgemeinschaft (DFG - German Research Council) passed the application for a 3rd funding period from 2020 to 2024, providing funding for the transregional research project to the sum of more than 7 million Euro over the next 4 years.

### Network of more than 100 researchers

The transregional research center is led by the TU Berlin as speaker university, with the Technical University München (TUM) as partner university. Furthermore, leading scientists from the FU Berlin, HU Berlin, Potsdam University, IST Austria, TU Wien and KAUST Saudi Arabia are involved as project leaders and researchers.

In more than 17 individual projects, over 100 researchers contribute their expertise to the enduring success of the project. A new development at the TUM is the involvement of professors and researchers from the Department of Informatics, who join the existing members of the Department of Mathematics to jointly address current problems on the cutting edge of both scientific fields.

### Discretizing geometric structures and processes

The SFB researches the discretization of geometric structures and dynamic processes, meaning, how to break down smooth geometric objects - such as curved surfaces - into simple basic building blocks or continually flowing dynamical processes - such as the flight paths of satellites - into discrete partial stages.

The common ground for this research in the field of geometry and dynamics is the search for and the study of discrete models which display properties and structures that are characteristic for the corresponding smooth geometric objects and dynamical processes.

The main aim is to see the discretization not only as an approximation of single phenomena in the continuation theory, but instead to discretize the complete underlying theory. In this way, the development of the discrete theory creates a new mathematical foundation, which includes the classical theory as a continuous limit under refinement of the discrete models.

### Solving a classical problem of differential geometry

These structure retaining discretizations were a focal part of the first two funding periods of the SFB. They led not only to the solving of important theoretical problems in mathematics, but also to the development of new applied methods in fields such as architecture, data analysis and computer graphics. For example, geometers at the TUM and IST Austria have developed a theory for the topological analysis of geometric data, which not only enables new perspectives in stochastic geometry, but also delivers practical methods for the processing and analysis of point clouds, such as the reconstruction of surfaces from laser scans.

Some application possibilities are surprising, such as the construction of non-reflective marginalized layers for the numerical treatment of wave phenomena. An example for an important theoretical result is the solving of a classical problem in differential geometry: the construction of specific examples show that there are closed surfaces which correspond in their internal metrics and central curvature without being congruent. All these results are based on structure retaining discretizations.

### SFB/TRR 109: focal areas in period 3

In the 3rd period of SFB, the scientists involved aim to bring geometry and dynamics closer together. Structure retaining discretizations therefore remain the central theme. The collaboration with members of the Department of Informatics at the TUM strengthen the approach towards applied research.

Fascinating themes for the final research period include:

- discrete conform models in geometry and mathematical physics
- rigidity of spin-systems
- topological data analysis
- structure retaining discretization of dynamical systems
- geometric structures in architecture

The promotion of talented young mathematicians through numerous research positions in the projects remains a central and important goal of the collaborative research center. Furthermore, the public relations work of the SFB, bringing more understanding for mathematical research to a general audience, is particularly supported through the production of documentary films explaining math research from a different perspective.

Further information of our department's involvement in the collaborative research center can be found under Discretization in Geometry and Dynamics.