**I** | **Mathematical challenges requiring software support** |

I.1 | Fourier matrices and character sheaves | Group representation theory |

I.2 | Generalised Gelfand-Graev representations in bad characteristic | Group representation theory |

I.3 | Imprimitive representations of quasisimple finite reductive groups | Representation theory, group theory |

I.4 | Decomposition matrices for exceptional groups of Lie type | Group representation theory |

I.5 | Derived categories of equivariant coherent sheaves | Algebraic geometry, constructive homological algebra, computer algebra |

I.6 | Syzygies and cohomology | Algebraic geometry, computer algebra |

I.7 | Construction of random points in moduli spaces and their geometry | Algebraic geometry, computer algebra |

I.8 | Algorithmic approaches to Teichmüller curves | Algebraic geometry |

I.9 | Moduli spaces of tropical varieties, modifications and compactifications | Tropical geometry |

I.10 | Tropical mirror symmetry of elliptic curves | Tropical geometry |

I.11 | Random matrices and Hurwitz numbers | Free probability, random matrices, tropical geometry |

I.12 | Noncommutative rational functions | Free probability, random matrices |

I.13 | Computational classification of orthogonal quantum groups | Operator algebras, compact quantum groups |

I.14 | Unit groups of orders | Algebraic number theory, arithmetic groups |

**II** | **Computational challenges posed by research in mathematics** |

II.1 | Central software project: Interaction, high-performance and support | Algorithmic number theory, group representation theory, polyhedral geometry, algebraic geometry, computer algebra |

II.2 | ANTIC: Explicit field construction | Algorithmic number theory |

II.3 | GAP: Generic character table of Spin$_8^+(q)$ | Group representation theory |

II.4 | POLYMAKE: polyhedral data, quotients and stacks | Polyhedral geometry |

II.5 | SINGULAR: A new level of abstraction and performance | Algebraic geometry, computer algebra |

II.6 | Gröbner bases and their applications over non-commutative rings | Non-commutative associative rings and algebras |

**III** | **Central projects** |

IRTG | Integrated Research Training Group | |

Z | Central tasks of the Collaborative Research Centre | |