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 aims at taking a leading role in meeting these challenges. The researchers within the TRR 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. The five core areas of the TRR, group and representation theory, algebraic geometry and commutative algebra, tropical and polyhedral geometry, non-commutative structures, number theory, are predestined for applying computer algebra methods. The TRR 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 are 1) to open up fundamental mathematical concepts to constructive treatment and design corresponding low- and high-level algorithms; 2) to attack and solve difficult mathematical problems, using algorithmic and experimental methods as key tools; 3) to support theoretical progress by constructing mathematical objects and generating databases and making them accessible to the mathematical community; 4) to design and further develop the visionary computer algebra system OSCAR for interdisciplinary research in the areas of the TRR and their application areas, implementing the new algorithms and integrating the databases there; 5) 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
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A01 - Character sheaves and generic character tables
(Project Heads
Geck, Meinolf
;
Lübeck, Frank
;
Malle, Gunter
)
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A05 - Decomposition matrices
(Project Head
Malle, Gunter
)
-
A10 - Algorithmic approaches to Teichmüller curves
(Project Heads
Bartholdi, Laurent
;
Weitze-Schmithüsen, Gabriela
)
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A11 - Linear degenerate flag varieties and their tropical counterparts
(Project Heads
Fang, Xin
;
Fourier, Ghislain
;
Markwig, Hannah
;
Nebe, Gabriele
)
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A13 - Plane Hurwitz numbers
(Project Heads
Agostini, Daniele
;
Böhm, Janko
;
Markwig, Hannah
)
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A17 - Generic orthogonal character tables
(Project Heads
Geck, Meinolf
;
Nebe, Gabriele
)
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A20 - Categorical representation theory
(Project Heads
Thiel, Ulrich
;
Weber, Moritz
)
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A22 - Effective Torelli Theorems
(Project Heads
Agostini, Daniele
;
Brandhorst, Simon
)
-
A23 - Algorithmic Minimal Model Program
(Project Heads
Böhm, Janko
;
Lazic, Vladimir
;
Schreyer, Frank-Olaf
;
Thiel, Ulrich
)
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A24 - Lattice polytopes and representation theory
(Project Heads
Fang, Xin
;
Fourier, Ghislain
;
Joswig, Michael
)
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A25 - Homomorphism counts for graphs and groups, classical and quantum
(Project Heads
Schweitzer, Pascal
;
Weber, Moritz
)
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A26 - Algorithms and logic for groups and dynamical systems
(Project Heads
Bartholdi, Laurent
;
Brandhorst, Simon
)
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B01 - Central software project: OSCAR
(Project Heads
Brandhorst, Simon
;
Decker, Wolfram
;
Fieker, Claus
;
Hofmann, Tommy
;
Horn, Max
;
Joswig, Michael
;
Lübeck, Frank
)
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B02 - Explicit class field theory in local fields
(Project Head
Fieker, Claus
)
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B04 - Computations in tropical combinatorics
(Project Heads
Joswig, Michael
;
Markwig, Hannah
;
Nebe, Gabriele
)
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B05 - Singular: Local-to-global Stuctures in Algebraic Geometry and Applications
(Project Heads
Brandhorst, Simon
;
Böhm, Janko
;
Decker, Wolfram
)
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B07 - Computations with matrix groups
(Project Heads
Horn, Max
;
Niemeyer, Alice
)
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B08 - Effective difference algebra and difference algebraic groups
(Project Heads
Bachmayr, Annette
;
Robertz, Daniel
)
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MGK - Integrated research training group
(Project Heads
Nebe, Gabriele
;
Speicher, Roland
;
Weber, Moritz
)
-
Z - Central tasks of the CRC 195
(Project Head
Malle, Gunter
)
Completed projects
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A02 - Generalised Gelfand-Graev representations, unipotent classes and nilpotent orbits
(Project Heads
Geck, Meinolf
;
Malle, Gunter
)
-
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
)
-
A14 - Topological recursion and free probability theory
(Project Head
Speicher, Roland
)
-
A15 - Noncommutative rational functions
(Project Head
Speicher, Roland
)
-
A16 - Computational classification of orthogonal quantum groups
(Project Head
Weber, Moritz
)
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A18 - Trivial source character tables of small finite groups
(Project Head
Lassueur, Caroline
)
-
A19 - Practical and theoretical aspects of the group isomorphism problem
(Project Head
Schweitzer, Pascal
)
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A21 - Arithmetic and Convexity in Buildings
(Project Heads
Nebe, Gabriele
;
Sturmfels, Ph.D., Bernd
)
-
B03 - GAP: Generic character table of Spin_8^+ (q)
(Project Heads
Lübeck, Frank
;
Malle, Gunter
)
-
B06 - Gröbner techniques for PBW deformations: parametrization, representations, applications
(Project Heads
Fourier, Ghislain
;
Thiel, Ulrich
;
Zerz, Eva
)
