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- 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.
- ItemTowards a Mathematical Theory of Turbulence in Fluids(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Bedrossian, JacobFluid mechanics is the theory of how liquids and gases move around. For the most part, the basic physics are well understood and the mathematical models look relatively simple. Despite this, fluids display a dazzling mystery to their motion. The random-looking, chaotic behavior of fluids is known as turbulence, and it lies far beyond our mathematical understanding, despite a century of intense research.
- 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.
- 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.
- ItemWie steuert man einen Kran?(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Altmann, Robert; Heiland, JanDie Steuerung einer Last an einem Kran ist ein technisch und mathematisch schwieriges Problem, da die Bewegung der Last nur indirekt beeinflusst werden kann. Anhand eines Masse-Feder-Systems illustrieren wir diese Schwierigkeiten und zeigen wie man mit einem zum konventionellen Lösungsweg alternativen Optimierungsansatz die auftretenden Komplikationen teilweise umgehen kann.
- ItemFokus-Erkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Deistler, Manfred; Graef, AndreasViele epileptische Anfälle entstehen in einer begrenzten Region im Gehirn, dem sogenannten Anfallsursprung. Eine chirurgische Entfernung dieser Region kann in vielen Fällen zu Anfallsfreiheit führen. Aus diesem Grund ist die Frage nach der Lokalisation des Anfallsursprungs aus EEG-Aufzeichnungen wichtig. Wir beschreiben hier ein Verfahren zur Lokalisation des Anfallsursprungs mittels Zeitreihenanalyse, das auf der Schätzung von Spektren im EEG beruht.
- 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.
- ItemRandom sampling of domino and lozenge tilings(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Fusy, ÉricA grid region is (roughly speaking) a collection of “elementary cells” (squares, for example, or triangles) in the plane. One can “tile” these grid regions by arranging the cells in pairs. In this snapshot we review different strategies to generate random tilings of large grid regions in the plane. This makes it possible to observe the behaviour of large random tilings, in particular the occurrence of boundary phenomena that have been the subject of intensive recent research.
- ItemHigh performance computing on smartphones(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Patera, Anthony T.; Urban, KarstenNowadays there is a strong demand to simulate even real-world engineering problems on small computing devices with very limited capacity, such as a smartphone. We explain, using a concrete example, how we can obtain a reduction in complexity – to enable such computations – using mathematical methods.