## Symbolic Tools in Mathematics and their Application

The Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) has established the transregional collaborative research centre (SFB-TRR) 195 “Symbolic Tools in Mathematics and their Application” starting in January 2017. The first funding period ended in 2020 with the second funding period of the SFB-TRR 195 having started in January 2021. Host university is the RPTU Kaiserslautern-Landau (RPTU Kaiserslautern), additional applicant universities are RWTH Aachen University and Saarland University.

Further participating researchers from TU Berlin, Max-Planck-Institute MiS Leipzig, University of Stuttgart, and University of Tübingen.

### Summary of the research programme

Computing examples has always been a key component of mathematical research. Modern computers paired with sophisticated mathematical software tools have taken the possibilities of such calculations to a new level. In the realm of algebra and its applications, where exact calculations are inevitable, the necessary software tools are provided by computer algebra systems. Current challenges in this area arise from the increasing complexity of examples, higher levels of abstraction and the need for interdisciplinary methods. The TRR 195 aims at taking a leading role in meeting these challenges.

The researchers within the TRR 195 have made pioneering contributions to computer algebra and rely on leading open source computer algebra systems developed (to a large extent) within the boundaries of the TRR 195.

The five core areas of the TRR 195,

- group and representation theory,
- algebraic geometry and commutative algebra,
- tropical and polyhedral geometry and
- non-commutative algebra and free probability theory and
- number theory,

are predestined for applying computer algebra methods. The TRR 195 offers the unique opportunity not only to guarantee further maintenance and development of these systems, but also to integrate them into a next generation computer algebra system, named OSCAR, providing interdisciplinary computational methods.

The principal contributions of the TRR 195 are

- to open up fundamental mathematical concepts to constructive treatment and design corresponding low- and high-level algorithms;
- to attack and solve difficult mathematical problems, using algorithmic and experimental methods as key tools;
- to support theoretical progress by constructing mathematical objects and generating databases and making them accessible to the mathematical community;
- to design and further develop the visionary computer algebra system OSCAR for interdisciplinary research in the areas of the TRR 195 and their application areas, implementing the new algorithms and integrating the databases there;
- to boost the performance of all components of OSCAR by combining new algorithms and technical advances, in particular through parallelization.