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- ItemEstimating the volume of a convex body(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Baldin, NicolaiSometimes 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.
- ItemSwallowtail on the shore(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Buchweitz, Ragnar-Olaf; Faber, EleonorePlatonic 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.
- ItemFrom Betti numbers to ℓ²-Betti numbers(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Kammeyer, Holger; Sauer, RomanWe provide a leisurely introduction to ℓ²-Betti numbers, which are topological invariants, by relating them to their much older cousins, Betti numbers. In the end we present an open research problem about ℓ²-Betti numbers.
- ItemConfiguration spaces and braid groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Jiménez Rolland, Rita; Xicoténcatl, Miguel A.In this snapshot we introduce configuration spaces and explain how a mathematician studies their ‘shape’. This will lead us to consider paths of configurations and braid groups, and to explore how algebraic properties of these groups determine features of the spaces.
- ItemFrom computer algorithms to quantum field theory: an introduction to operads(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Krähmer, UlrichAn 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.
- ItemDas Problem der Kugelpackung(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Dostert, Maria; Krupp, Stefan; Rolfes, Jan HendrikWie 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 Dimensionen schwierig.
- ItemZopfgruppen, die Yang–Baxter-Gleichung und Unterfaktoren(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Lechner, GandalfDie Yang–Baxter-Gleichung ist eine faszinierende Gleichung, die in vielen Gebieten der Physik und der Mathematik auftritt und die am besten diagrammatisch dargestellt wird. Dieser Snapshot schlägt einen weiten Bogen vom Zöpfeflechten über die Yang–Baxter- Gleichung bis hin zur aktuellen Forschung zu Systemen von unendlichdimensionalen Algebren, die wir „Unterfaktoren“ nennen.
- ItemLagrangian mean curvature flow(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Lotay, Jason D.Lagrangian mean curvature flow is a powerful tool in modern mathematics with connections to topics in analysis, geometry, topology and mathematical physics. I will describe some of the key aspects of Lagrangian mean curvature flow, some recent progress, and some major open problems.
- ItemPositive Scalar Curvature and Applications(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Rosenberg, Jonathan; Wraith, DavidWe introduce the idea of curvature, including how it developed historically, and focus on the scalar curvature of a manifold. A major current research topic involves understanding positive scalar curvature. We discuss why this is interesting and how it relates to general relativity.
- ItemClosed geodesics on surfaces and Riemannian manifolds(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Radeschi, MarcoGeodesics are special paths in surfaces and so-called Riemannian manifolds which connect close points in the shortest way. Closed geodesics are geodesics which go back to where they started. In this snapshot we talk about these special paths, and the efforts to find closed geodesics.