Intro -- Preface -- Large hyperbolic polygons and hyperelliptic Riemann surfaces -- 1. Hyperbolic Polygons -- 2. Closing large polygons -- 3. Groups from Polygons -- 4. Reduction Algorithms on Polygons -- References -- On isolated strata of pentagonal Riemann surfaces in the branch locus of moduli spaces -- 1. Introduction -- 2. Riemann surfaces and Fuchsian groups -- 3. Disconnectedness by pentagonal Riemann surfaces -- References -- Finite group actions of large order on compact bordered surfaces -- 1. Introduction -- 2. Preliminaries -- 3. Signatures associated with groups of order ≥6( -1) -- 4. A computational approach -- 5. Finite group actions of order 12( -1) -- 6. Finite group actions of order 8( -1) -- 7. Finite group actions of order (20)/3( -1) -- 8. Finite group actions of order 6( -1) -- 9. Infinite families -- References -- Surfaces of low degree containing a canonical curve -- 1. Introduction -- 2. The background on ruled surfaces and Δ-genus -- 3. Minimal degree surfaces containing a canonical curve -- 4. Low degree examples -- References -- Ideals of curves given by points -- 1. Introduction -- 2. Border bases for homogeneous ideals -- 3. Curves given by points -- 4. Curves in ℙⁿ(ℂ) given by approximate points -- 5. Degree bounds for ideal generators -- Acknowledgments -- References -- Non-genera of curves with automorphisms in characteristic -- 1. Introduction -- 2. Non-genera for ℤ/ ℤ-actions -- 3. -ranks -- References -- Numerical Schottky uniformizations of certain cyclic L-gonal curves -- 1. Introduction -- 2. Preliminaries -- 3. Basic facts on Whittaker groups -- 4. Basic objects -- 5. The generalized Myrberg's algorithm -- 6. Application to algebraic curves -- Acknowledgements -- References -- Generalized lantern relations and planar line arrangements -- 1. Introduction. 2. Real line arrangements and relations on Dehn twists -- 3. Applications -- References -- Effective -adic cohomology for cyclic cubic threefolds -- 1. Zeta functions: generalities -- 2. Algebraic de Rham cohomology -- 3. de Rham cohomology and -adic cohomology -- 4. The direct method for cyclic cubic threefolds -- 5. Picard-Fuchs-Manin connections -- 6. The deformation method for cyclic cubic threefolds -- Appendix A. Notes and further reading -- References -- Generating sets of affine groups of low genus -- 1. Introduction -- 2. The Algorithm -- 3. Genus Zero Systems and the Guralnick-Thompson Conjecture -- References -- Classification of algebraic ODEs with respect to rational solvability -- 1. Introduction -- 2. Preliminaries -- 3. A group of affine linear transformations -- 4. Solvable AODEs and their associated systems -- 5. Parametrizable ODEs with special geometric shapes -- 6. Conclusion -- References -- Circle packings on conformal and affine tori -- 1. Introduction -- 2. Tori -- 3. Circle Packings -- 4. Packing Labels for Combinatorial Tori: New Existence Proof -- 5. Packings on Affine Tori -- References -- Effective radical parametrization of trigonal curves -- 1. Introduction -- 2. Classical results on trigonality -- 3. Lie algebras -- 4. The Lie algebra method for trigonal curves -- 5. Computational experiences -- References.