TRR 195:  Symbolic Tools in Mathematics and their Application

• Subject Area: Mathematics
• Term: since 2017
• Website: Homepage
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 286237555

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, non-commutative algebra and free probability theory, number theory, are predestined for apply-ing 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 da-tabases there;• to boost the performance of all components of OSCAR by combining new algorithms and technical advances, in particular through parallelization.

• DFG Programme: CRC/Transregios

Current projects

• A01 - Fourier matrices and character sheaves (Project Heads Geck, Meinolf ;  Lübeck, Frank )
• A02 - Generalised Gelfand-Graev representations, unipotent classes and nilpotent orbits (Project Heads Geck, Meinolf ;  Malle, Gunter )
• A05 - Decomposition matrices (Project Head Malle, Gunter )
• A10 - Algorithmic approaches to Teichmüller curves (Project Head Weitze-Schmithüsen, Gabriela )
• A11 - Tropical linear degenerate flag varieties and applications (Project Heads Fourier, Ghislain ;  Markwig, Hannah ;  Nebe, Gabriele )
• A13 - Generating series for curve counting (Project Heads Böhm, Janko ;  Markwig, Hannah )
• A14 - Topological recursion and free probability theory (Project Head Speicher, Roland )
• A17 - Clifford orders (Project Head Nebe, Gabriele )
• A19 - Practical and theoretical aspects of the group isomorphism problem (Project Head Schweitzer, Pascal )
• A20 - Towards unipotent character sheaves associated to Coxeter groups (Project Head Thiel, Ulrich )
• A21 - Arithmetic and Convexity in Buildings (Project Heads Nebe, Gabriele ;  Sturmfels, Ph.D., Bernd )
• A22 - Automorphisms of irreducible holomorphic symplectic manifolds via hermitian forms (Project Head Brandhorst, Simon )
• A23 - Conjectures and new examples in birational geometry (Project Heads Lazic, Vladimir ;  Schreyer, Frank-Olaf ;  Thiel, Ulrich )
• A24 - Lattice polytopes, partially ordered sets, spherical varieties (Project Heads Fourier, Ghislain ;  Joswig, Michael )
• A25 - Quantum symmetries and quantum isomorphisms of graphs (Project Head Weber, Moritz )
• A26 - Algorithms and Logic for automata groups (Project Head Bartholdi, Laurent )
• B01 - Central software project: OSCAR (Project Heads Decker, Wolfram ;  Fieker, Claus ;  Horn, Max ;  Joswig, Michael ;  Lübeck, Frank )
• B02 - Galois cohomology for non-normal fields (Project Head Fieker, Claus )
• B04 - Enumerating tropical curves and hypersurfaces (Project Heads Joswig, Michael ;  Markwig, Hannah )
• B05 - SINGULAR: - Massively parallel structures in algebraic geometry (Project Heads Brandhorst, Simon ;  Böhm, Janko ;  Decker, Wolfram )
• B06 - Gröbner techniques for PBW deformations: parametrization, representations, applications (Project Heads Fourier, Ghislain ;  Thiel, Ulrich ;  Zerz, Eva )
• B07 - Computations with matrix groups (Project Heads Horn, Max ;  Niemeyer, Alice )
• MGK - Integrated Research Training Group (Project Heads Nebe, Gabriele ;  Speicher, Roland )
• Z - Central tasks of the Collaborative Research Centre (Project Head Malle, Gunter )

Completed projects

• A03 - Imprimitive representations of quasisimple finite reductive groups (Project Head Hiß, Gerhard )
• A07 - Derived categories of equivariant coherent sheaves (Project Heads Barakat, Mohamed ;  Schreyer, Frank-Olaf )
• A08 - Syzygies and cohomology (Project Head Schreyer, Frank-Olaf )
• A09 - Construction of random points in moduli spaces and their geometry (Project Heads Decker, Wolfram ;  Schreyer, Frank-Olaf )
• A15 - Noncommutative rational functions (Project Head Speicher, Roland )
• A16 - Computational classification of orthogonal quantum groups (Project Head Weber, Moritz )
• A18 - Trivial source character tables of small finite groups (Project Head Lassueur, Caroline )
• B03 - GAP: Generic character table of Spin_8^+ (q) (Project Heads Lübeck, Frank ;  Malle, Gunter )

• Applicant Institution: Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau
• Co-Applicant Institution: Rheinisch-Westfälische Technische Hochschule Aachen; Universität des Saarlandes
• Participating University: Eberhard Karls Universität Tübingen; Technische Universität Berlin; Universität Stuttgart
• Participating Institution: Max-Planck-Institut für Mathematik in den Naturwissenschaften (MIS)
• Spokesperson: Professor Dr. Gunter Malle