Gebhard Böckle is professor of mathematics at the Universität Heidelberg. His research themes are Galois representations over number and function fields, the arithmetic of function fields, and cohomological methods in positive characteristic.Wolfram Decker is professor of mathematics at TU Kaiserslautern. His research fields are algebraic geometry and computer algebra. He heads the development team of the computer algebra system Singular. From 2010-2016, he was the coordinator of the DFG Priority Program SPP 1489 from which this volume originates.Gunter Malle is professor of mathematics at TU Kaiserslautern. He is working in group representation theory with particular emphasis on algorithmic aspects.

• Introduction.- Bächle et al: Algorithmic aspects of units in group rings.- M. Barakat et al: A constructice approach to the module of twisted glocal sections on relative projective spaces.- J. Böhm et al: Local to global algorithms for the Gorenstein adjoint ideal of a curve.- M. Börner et al: Picard curves with small conductor.- W. Bruns et al: Normaliz 2013-2016.- T. Centeleghe et al: Integral Frobenius for abelian varieties with real multiplication.- M. Dettweiler et al: Monodromy of the multiplicative and the additive convolution.- B. Eick et al: Constructing groups of ‘small’ order: Recent results and open problems.- B. Eick et al: Classifying nilpotent associative algebras: small coclass and finite fields.- A. Fruehbis-Krüger et al: Desingularization of arithmetic surfaces: algorithmic aspects.-  A. Gathmann et al: Moduli spaces of curves in tropical varieties.- A. Gathmann et al: Tropical moduli spaces of stable maps to a curve.- M. Geck et al: Invariant bilinear forms on W-graph representations and linear algebra over integral domains.- S. Hampe et al: Tropical computations in polymake.- M. Hoff: Focal schemes to families of secant spaces to canonical curves.- T. Hoge et al: Inductive and recursive freeness of localizations of multiarrangements.- L. Kastner: Toric ext and tor in polymake and Singular: The twodimensional case and beyond.- M. Lange-Hegermann et al: The differential dimension polynomial for characterizable differential ideals.- V. Levandovskyy: Factorization of Z-homogeneous polynomials in the first q-Weyl algebra.- E.W. Mayr et al: Complexity of membership problems of different types of polynomial ideals.- T. Moeller et al: Localizations of inductively factored arrangements.- G. Nebe et al: One class genera of lattice chains over number fields.- A. Paffenholz: polyDB: A database for polytopes and related objects.- G. G. Pfister et al: Construction of neron desingularization for two-dimensional rings. - T. Rossmann et al: A framework for computing zeta functions of groups, algebras, and modules.- A. Shalile: On decomposition numbers of diagram algebras.- U. Spreckels et al: Koblitz’ conjecture for abelian varieties.- M. Stoll: Chabauty without the Mordell-Weil group.- M. Stoll: An explicit theory of heights for hyperelliptic Jacobians of genus three.- T. Theobald: Some recent developments in spectrahedral computation.- G. Wiese et al: Topics on modular Galois representations modulo prime powers.