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Fourth Power Moment of Dedekind Zeta-Functions of Real Quadratic Number Fields with Class Number One
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In Memoriam: Hans Duistermaat (1942-2010) : grasping the essence
No full textOn March 19, 2010, mathematics lost one of its leading geometric analysts, Johannes Jisse Duistermaat. At age 67 he passed away, after a short illness following a renewed bout of lymphoma the doctors thought they had controlled. ‘Hans’, as Duistermaat was universally known among friends and colleagues, was not only a brilliant research mathematician and inspiring teacher, but also an accomplished chess player, very fond of several physical sports, and a devoted husband and (grand)father. Erik van den Ban and Johan Kolk look back on Duistermaat’s life and work in this contribution; it is followed by remembrances and surveys by some of his friends, students, and colleague
A dialogue on the use of arithmetic in geometry: Van Ceulen’s and Snellius’s Fundamenta Arithmetica et Geometrica
No full textSnellius’s Fundamenta Arithmetica et Geometrica (1615) is much more than a Latin translation of Ludolph van Ceulen’s Arithmetische en Geometrische Fondamenten. Willebrord Snellius both adapted and commented on the Dutch original in his Fundamenta, and thus his Latin version can be read as a dialogue between representatives of two different approaches to mathematics in the early modern period: Snellius’s humanist approach and Van Ceulen’s practitioner’s approach. This article considers the relationship between the Dutch and Latin versions of the text and, in particular, puts some of their statements on the use of numbers in geometry under the microscope. In addition, Snellius’s use of the Fundamenta as an instrument to further his career is explained
Measure-theoretic rigidity for Mumford curves
No full textOne can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a non- Archimedean analogue of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson– Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain non-Archimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage, Enriques and Petri on equations for the canonical embedding of a curve
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