The transparency and power of geometric constructions has been a source of inspiration to generations of mathematicians. The beauty and persuasion of pictures, communicated in words or drawings, continues to provide the intuition and arguments for working with complicated concepts and structures of modern mathematics. This volume contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory. Key topics include: - Curves and their Jacobians - Algebraic surfaceModuli spaces, Shimura varieties - Motives and motivic integration - Number-theoretic applications, rational points - Combinatorial aspects of algebraic geometry - Quantum cohomology - Arithmetic dynamical systems The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry; the text can serve as an intense introduction for graduate students and those wishing to pursue research in these areas.

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