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- ItemTropical geometry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Brugallé, Erwan; Itenberg, Ilia; Shaw, Kristin; Viro, OlegWhat 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.
- ItemFootballs and donuts in four dimensions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Klee, StevenIn 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.
- ItemFrom the dollar game to the Riemann-Roch Theorem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Lamboglia, Sara; Ulirsch, MartinWhat is the dollar game? What can you do to win it? Can you always win it? In this snapshot you will find answers to these questions as well as several of the mathematical surprises that lurk in the background, including a new perspective on a century-old theorem.
- 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.
- ItemAperiodic Order and Spectral Properties(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Baake, Michael; Damanik, David; Grimm, UwePeriodic structures like a typical tiled kitchen floor or the arrangement of carbon atoms in a diamond crystal certainly possess a high degree of order. But what is order without periodicity? In this snapshot, we are going to explore highly ordered structures that are substantially nonperiodic, or aperiodic. As we construct such structures, we will discover surprising connections to various branches of mathematics, materials science, and physics. Let us catch a glimpse into the inherent beauty of aperiodic order!
- ItemFinite geometries: pure mathematics close to applications(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Storme, LeoThe research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”.
- ItemWinkeltreue zahlt sich aus(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Günther, FelixNicht nur Seefahrerinnen, auch Computergrafikerinnen und Physikerinnen wissen Winkeltreue zu schätzen. Doch beschränkte Rechenkapazitäten und Vereinfachungen in theoretischen Modellen erfordern es, winkeltreue Abbildungen nur mit einer überschaubaren Datenmenge zu beschreiben. Entsprechende Theorien werden in der diskreten Mathematik untersucht. Im Folgenden lade ich Sie auf eine Reise in die faszinierende Welt der winkeltreuen Abbildungen ein.