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- ItemQuantum diffusion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Knowles, AnttiIf you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see something that is never observed in the real world. Such diffusive and irreversible behaviour is ubiquitous in nature. Nevertheless, the fundamental equations that describe the motion of individual particles – Newton’s and Schrödinger’s equations – are reversible in time: a film depicting the motion of just a few particles looks as realistic when played forwards as when played backwards. In this snapshot, we discuss how one may try to understand the origin of diffusion starting from the fundamental laws of quantum mechanics.
- 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.
- ItemOperator theory and the singular value decomposition(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Knese, GregThis is a snapshot about operator theory and one of its fundamental tools: the singular value decomposition (SVD). The SVD breaks up linear transformations into simpler mappings, thus unveiling their geometric properties. This tool has become important in many areas of applied mathematics for its ability to organize information. We discuss the SVD in the concrete situation of linear transformations of the plane (such as rotations, reflections, etc.).
- ItemTouching the transcendentals: tractional motion from the bir th of calculus to future perspectives(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Milici, PietroWhen the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century but today, mainstream conception relegates geometry to be merely a tool of visualization. In this snapshot, however, we consider geometric and constructive components of calculus. We reinterpret “tractional motion”, a late 17th century method to draw transcendental curves, in order to reintroduce “ideal machines” in math foundation for a constructive approach to calculus that avoids the concept of infinity.
- 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.
- ItemExpander graphs and where to find them(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Khukhro, AnaGraphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours.
- ItemOn Logic, Choices and Games(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Oliva, PauloCan we always mathematically formalise our taste and preferences? We discuss how this has been done historically in the field of game theory, and how recent ideas from logic and computer science have brought an interesting twist to this beautiful theory.
- 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!
- ItemMathematische Modellierung von Krebswachstum(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Engwer, Christian; Knappitsch, MarkusKrebs ist eine der größten Herausforderungen der modernen Medizin. Der WHO zufolge starben 2012 weltweit 8,2 Millionen Menschen an Krebs. Bis heute sind dessen molekulare Mechanismen nur in Teilen verstanden, was eine erfolgreiche Behandlung erschwert. Mathematische Modellierung und Computersimulationen können helfen, die Mechanismen des Tumorwachstums besser zu verstehen. Sie eröffnen somit neue Chancen für zukünftige Behandlungsmethoden. In diesem Schnappschuss steht die mathematische Modellierung von Glioblastomen im Fokus, einer Klasse sehr agressiver Tumore im menschlichen Gehirn.
- ItemThe Mathematics of Fluids and Solids(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Kaltenbacher, Barbara; Kukavica, Igor; Lasiecka, Irena; Triggiani, Roberto; Tuffaha, Amjad; Webster, JustinFluid-structure interaction is a rich and active field of mathematics that studies the interaction between fluids and solid objects. In this short article, we give a glimpse into this exciting field, as well as a sample of the most significant questions that mathematicians try to answer.
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