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Focuses on the interaction between algebra and algebraic geometry, including high-level research papers and surveys contributed by over 40 top specialists representing more than 15 countries worldwide. Describes abelian groups and lattices, algebras and binomial ideals, cones and fans, affine and projective algebraic varieties, simplicial and cellular complexes, polytopes, and arithmetics. Preface Frobenius and Maschke Type Theorems for Doi- Hopf Modules and Entwined Modules Revisited: A Unified Approach Computing the Gelfand-Kirillov Dimension II Some Problems About Nilpotent Lie Algebras On L*-Triples and Jordan H*-Pairs Toric Mathematics from Semigroup Viewpoint Canonical Forms for Linear Dynamical Systems over Commutative Rings: The Local Case An introduction to Janet bases and Grobner bases Invariants of Coalgebras Multiplication Objects Krull-Schmidt Theorem and Semilocal Endomorphism Rings On Suslin's Stability Theorem for R[XI...xm] Characterization of Rings Using Socle-Fine and Radical-Fine Notions About Bernstein Algebras About an Algorithm of T. Oaku Minimal Injective Resolutions: Old and New Special Divisors of Blowup Algebras Existence of Euler Vector Fields for Curves with Binomial Ideal An Amitsur Cohomology Exact Sequence for Invo-lutive Brauer Groups of the Second Kind 1 Computation of the Slopes of a D-Module of Type Dr/N Symmetric Closed Categories and Involutive Brauer Groups1