34 research outputs found

    Two-dimensional semi-log-canonical hypersurfaces

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    We derive explicit equations for all two-dimensional, semi-log-canonical hypersurface singularities by an elemetary method

    Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable

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    π₁ of Miranda moduli spaces of elliptic surfaces

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    We give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over [Formula: see text] which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of [Formula: see text] presented in terms of [Formula: see text] , [Formula: see text] on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other. Our approach exploits the natural [Formula: see text] -action on Weierstrass curves and the identification of [Formula: see text] -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over [Formula: see text] . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition

    Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere

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    We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system. It yields an explicit symplectic representation of the braid groups (coloured or not) of four strings

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    History of Mathematics: Models and Visualization in the Mathematical and Physical Sciences

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    This workshop brought together historians of mathematics and science as well as mathematicians to explore important historical developments connected with models and visual elements in the mathematical and physical sciences. It addressed the larger question of what has been meant by a model, a notion that has seldom been subjected to careful historical study. Most of the talks dealt with case studies from the period 1800 to 1950 that covered a number of analytical, geometrical, mechanical, astronomical, and physical phenomena. The workshop also considered the role of visual thinking as a component of mathematical creativity and understanding
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