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- ItemSolving quadratic equations in many variables(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Tignol, Jean-PierreFields are number systems in which every linear equation has a solution, such as the set of all rational numbers Q or the set of all real numbers R. All fields have the same properties in relation with systems of linear equations, but quadratic equations behave differently from field to field. Is there a field in which every quadratic equation in five variables has a solution, but some quadratic equation in four variables has no solution? The answer is in this snapshot.
- ItemA few shades of interpolation(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Szpond, JustynaThe 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.
- ItemNews on quadratic polynomials(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Pottmeyer, LukasMany 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.
- ItemPrime tuples in function fields(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Bary-Soroker, LiorHow many prime numbers are there? How are they distributed among other numbers? These are questions that have intrigued mathematicians since ancient times. However, many questions in this area have remained unsolved, and seemingly unsolvable in the forseeable future. In this snapshot, we will discuss one such problem, the Twin Prime Conjecture, and a quantitative version of it known as the Hardy–Littlewood Conjecture. We will also see that these and other questions about prime numbers can be extended to questions about function fields, and discuss recent progress which has been made to answer them in this context.
- 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.
- ItemSymmetry and characters of finite groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Giannelli, Eugenio; Taylor, JayOver the last two centuries mathematicians have developed an elegant abstract framework to study the natural idea of symmetry. The aim of this snapshot is to gently guide the interested reader through these ideas. In particular, we introduce finite groups and their representations and try to indicate their central role in understanding symmetry.
- 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!
- ItemOn the containment problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Szemberg, Tomasz; Szpond, JustynaMathematicians routinely speak two languages: the language of geometry and the language of algebra. When translating between these languages, curves and lines become sets of polynomials called “ideals”. Often there are several possible translations. Then the mystery is how these possible translations relate to each other. We present how geometry itself gives insights into this question.
- ItemPolyhedra and commensurability(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Guglielmetti, Rafael; Jacquement, MatthieuThis 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.
- ItemProfinite groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Bartholdi, LaurentProfinite 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.