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End date: 2019 × Start date: 2010 × Type: report × Subject: Algebra and Number Theory ×

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Now showing 1 - 10 of 29
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    Polyhedra and commensurability
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Guglielmetti, Rafael; Jacquement, Matthieu
    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 intriguing connections with other fields of mathematics.
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    Arrangements of lines
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Harbourne, Brian; Szemberg, Tomasz
    We discuss certain open problems in the context of arrangements of lines in the plane.
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    Profinite groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Bartholdi, Laurent
    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.
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    Computing with symmetries
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Roney-Dougal, Colva M.
    Group theory is the study of symmetry, and has many applications both within and outside mathematics. In this snapshot, we give a brief introduction to symmetries, and how to compute with them.
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    A few shades of interpolation
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Szpond, Justyna
    The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points from given data points. The new points usually lie in between the already-known points. The purpose of this snapshot is to introduce a particular type of interpolation, namely, polynomial interpolation. This will be explained starting from basic ideas that go back to the ancient Babylonians and Greeks, and will arrive at subjects of current research activity.
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    Snake graphs, perfect matchings and continued fractions
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Schiffler, Ralf
    A continued fraction is a way of representing a real number by a sequence of integers. We present a new way to think about these continued fractions using snake graphs, which are sequences of squares in the plane. You start with one square, add another to the right or to the top, then another to the right or the top of the previous one, and so on. Each continued fraction corresponds to a snake graph and vice versa, via “perfect matchings” of the snake graph. We explain what this means and why a mathematician would call this a combinatorial realization of continued fractions.
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    News on quadratic polynomials
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Pottmeyer, Lukas
    Many problems in mathematics have remained unsolved because of missing links between mathematical disciplines, such as algebra, geometry, analysis, or number theory. Here we introduce a recently discovered result concerning quadratic polynomials, which uses a bridge between algebra and analysis. We study the iterations of quadratic polynomials, obtained by computing the value of a polynomial for a given number and feeding the outcome into the exact same polynomial again. These iterations of polynomials have interesting applications, such as in fractal theory.
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    Expander graphs and where to find them
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Khukhro, Ana
    Graphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours.
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    Ideas of Newton-Okounkov bodies
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Kiritchenko, Valentina; Timorin, Vladlen; Smirnov, Evgeny
    In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion.
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    Swallowtail on the shore
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Buchweitz, Ragnar-Olaf; Faber, Eleonore
    Platonic solids, Felix Klein, H.S.M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not only for their sheer beauty but also because of their many symmetry properties. In this snapshot we will start from these symmetries, move on to groups, singularities, and finally find the connection between a tetrahedron and a “swallowtail”. Our running example is the tetrahedron, but every construction can be carried out with any other of the Platonic solids.