## 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. Host university is the University of Kaiserslautern (TU Kaiserslautern), additional applicant universities are RWTH Aachen University and Saarland University.

Further participating researchers from TU Berlin, University of Siegen, University of Stuttgart, and University of Tübingen.

### Summary of the research programme

Experiments based on calculating examples have always played a key role in mathematical research. Modern computers paired with sophisticated mathematical software tools allow for far reaching experiments which were previously unimaginable. They enable mathematicians to test working hypotheses or conjectures in a large number of instances; to find counterexamples or enough mathematical evidence to sharpen a conjecture; to arrive at new conjectures in the first place; to verify theorems whose proofs have been reduced to handling a finite number of special cases.

In the realm of algebra and its applications, where exact calculations are inevitable, the desired software tools are provided by computer algebra systems. These systems are large, complex pieces of software, containing and relying on a vast amount of mathematical reasoning. Driven by intended applications, they are created by collaborative efforts requiring specialists in many different fields. It is an important aspect that through these systems a large treasure of mathematical knowledge becomes accessible to and can also be applied by non-experts.

A decisive feature of current developments is that more and more of the abstract concepts of pure mathematics are made constructive, with interdisciplinary methods playing a significant role. The TRR 195 aims at taking a leading role in driving these developments: In the five core areas listed below, it will provide the computational open source infrastructure for years to come; it will create vast amounts of data important to the mathematical community; and it will exploit the infrastructure and data to solve fundamental mathematical problems.

Four of the five core areas of the TRR 195,

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

are predestined for applying computer algebra methods, with pioneering work of the participating researchers and institutions on all levels — providing computational access to mathematical concepts, designing algorithms, implementing them and applying them to profound mathematical questions. As a distinctive feature, the TRR 195 can rely on leading open source computer algebra systems developed (to a large extent) within its boundaries, and addressing all of the above core areas.

In fact, the respective systems provide indispensable tools for thousands of researchers worldwide, and the TRR 195 offers the unique opportunity not only to guarantee their maintenance and further development, but also to integrate the systems and additional libraries and packages into a unified next generation open source computer algebra system providing interdisciplinary computational methods.

In its fifth core area,

- random matrix theory and free probability,

the TRR 195 will bundle ground-breaking work on the theoretical side with the combined expertise of the TRR 195 on the practical side, making this core area accessible to an algebraic and computational treatment, while simultaneously exploring the interface of combinatorics, algebraic and tropical geometry, random matrix theory and free probability in mathematical research.

The principal contributions of the TRR 195 will be

- 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 integrate the systems, libraries and packages developed within the TRR 195 into a unified computer algebra system which supports interdisciplinary research in the areas of the TRR 195, implementing the new algorithms and integrating the databases there;
- to enhance the performance of all components of the unified system, in particular by designing and implementing parallel algorithms for the efficient use of multicore computers and high-performance clusters.