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Has files: Yes × Type: report × Subject: Geometry and Topology ×

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Now showing 1 - 10 of 40
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    Jewellery from tessellations of hyperbolic space
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Gangl, Herbert
    In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.
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    Estimating the volume of a convex body
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Baldin, Nicolai
    Sometimes the volume of a convex body needs to be estimated, if we cannot calculate it analytically. We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape.
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    Tropical geometry
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Brugallé, Erwan; Itenberg, Ilia; Shaw, Kristin; Viro, Oleg
    What kind of strange spaces hide behind the enigmatic name of tropical geometry? In the tropics, just as in other geometries, one of the simplest objects is a line. Therefore, we begin our exploration by considering tropical lines. Afterwards, we take a look at tropical arithmetic and algebra, and describe how to define tropical curves using tropical polynomials.
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    Describing distance: from the plane to spectral triples
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Arici, Francesca; Mesland, Bram
    Geometry draws its power from the abstract structures that govern the shapes found in the real world. These abstractions often provide deeper insights into the underlying mathematical objects. In this snapshot, we give a glimpse into how certain “curved spaces” called manifolds can be better understood by looking at the (complex) differentiable functions they admit.
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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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    Footballs and donuts in four dimensions
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Klee, Steven
    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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    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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    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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    Mixed volumes and mixed integrals
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Rotem, Liran
    In recent years, mathematicians have developed new approaches to study convex sets: instead of considering convex sets themselves, they explore certain functions or measures that are related to them. Problems from convex geometry become thereby accessible to analytic and probabilistic tools, and we can use these tools to make progress on very difficult open problems. We discuss in this Snapshot such a functional extension of some “volumes” which measure how “big” a set is. We recall the construction of “intrinsic volumes”, discuss the fundamental inequalities between them, and explain the functional extensions of these results.