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Profinite objects are mathematical constructions used to collect, in a uniform manner, facts about infinitely many finite objects. We shall review recent progress in the theory of profinite groups, due to Nikolov and Segal, and its implications for finite groups.
Submitted by Antonia Mey on
In this snapshot, we explore connections between the mathematical areas of counting and geometry by studying objects called simplicial complexes. We begin by exploring many familiar objects in our three dimensional world and then discuss the ways one may generalize these ideas into higher dimensions.
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The Willmore problem studies which torus has the least amount of bending energy. We explain how to think of a torus as a donut-shaped surface and how the intuitive notion of bending has been studied by mathematics over time.
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This snapshot introduces the notion of commensurability of polyhedra. At its bottom, this concept can be developed from constructions with paper, scissors, and glue. Starting with an elementary example, we formalize it subsequently. Finally, we discuss intrigu- ing connections with other fields of mathematics.
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Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3-dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen Kommunikation hat, hört sich einfach an, ist jedoch für Kugeln in höheren Dimensionen noch immer ungelöst. Sogar die Berechnung guter Näherungslösungen ist für die meisten Di- mensionen schwierig.
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An operad is an abstract mathematical tool encoding operations on specific mathematical structures. It finds applications in many areas of mathematics and related fields. This snapshot explains the concept of an operad and of an algebra over an operad, with a view towards a conjecture formulated by the mathematician Pierre Deligne. Deligne’s (by now proven) conjecture also gives deep inights into mathematical physics.
Submitted by Antonia Mey on
Anlässlich des internationalen Jahres der Kristallografie wurde ein Artikel aus dem Italienischen ins Deutsche für IMAGINARY übersetzt. Es geht um verschiedene Kristallstrukturen und wie man ursprünglich auf Kristallstrukturen gestoßen ist.
Der orginal Artikel ist in der Zeitschrift XlaTangente am 1. Oktober 2014 verköffentlicht worden. Hier, geht es zum Orginal.
Submitted by Bianca Violet on
Was haben das Mineral Pyrit, der fünfte platonische K̈örper (genannt Dodekaeder) und ein Polynom vom Grad 16 gemeinsam? Dieser Artikel untersucht diesen Zusammenhang mit Hilfe der kostenlosen Software SURFER der Plattform IMAGINARY - open mathematics. Daraus entstehen faszinierende Bilder, die zeigen, wie ein Würfel sich nacheinander in einen Dodekaeder, einen Rhombendodekaeder und einen Oktaeder verwandelt, alles dank einer einzigen Formel. Es wird ein Überblick über die Ideen und die Mathematik hinter diesen Visualisierungen gegeben.
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Humans have been fascinated by crystals for a long time. Its regular geometry, its special symmetry, but also its diversity in colours surprise and please us. In this article, an overview of the connection between crystals and mathematics is given. It is a contribution to the International Year of Crystallography 2014.
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This snapshot is about zero-dimensional symmetry. Thanks to recent discoveries we now understand such symmetry better than previously imagined possible. While still far from complete, a picture of zero-dimensional symmetry is beginning to emerge.