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Research of the Algorithmic Algebra Group Galois theory ( Max Albert, Sebastian Basten, Michael Dettweiler, Michael Maier, Prof. B. Heinrich Matzat): Inverse Galois theory. Galois representations. Computation of Galois groups. Construction of polynomials with given Galois groups. Differential Galois theory (Michael Dettweiler, Julia Hartmann, Prof. B. Heinrich Matzat, Andreas Maurischat, Michael Wibmer): Inverse problems and embedding problems. p-adic differential equations. Modular differential Galois theory. Computation of differential Galois groups. Construction of differential equations with given Galois group. Invariant theory ( Emilie Dufresne, Julia Hartmann, Prof. B. Heinrich Matzat, Andreas Maurischat): Invariant theory of finite groups. Modular invariant theory. Noether's problem and generic polynomials. Applications of invariant theory. Separating invariants. Computeralgebra / Software development (Marcus Hausdorf, Prof. B. Heinrich Matzat, T. Olschewski, Werner Seiler): Computational algebraic number theory. Involutive basis. Computational invariant theory: INVAR: A software package for working with polynomial invariants of finite groups. The invariant package in MAGMA Databases for differential equations with given differential Galois group.