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- 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.
- ItemC∗ -algebras: structure and classification(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Kerr, DavidThe theory of C∗C∗-algebras traces its origins back to the development of quantum mechanics and it has evolved into a large and highly active field of mathematics. Much of the progress over the last couple of decades has been driven by an ambitious program of classification launched by George A. Elliott in the 1980s, and just recently this project has succeeded in achieving one of its central goals in an unexpectedly dramatic fashion. This Snapshot aims to recount some of the fundamental ideas at play.
- ItemMolecular Quantum Dynamics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Hagedorn, George A.; Lasser, CarolineWe provide a brief introduction to some basic ideas of Molecular Quantum Dynamics. We discuss the scope, strengths and main applications of this field of science. Finally, we also mention open problems of current interest in this exciting subject.
- ItemQuantum symmetry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Weber, MoritzIn mathematics, symmetry is usually captured using the formalism of groups. However, the developments of the past few decades revealed the need to go beyond groups: to “quantum groups”. We explain the passage from spaces to quantum spaces, from groups to quantum groups, and from symmetry to quantum symmetry, following an analytical approach.
- ItemDarcy's law and groundwater flow modelling(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Schweizer, BenFormulations of natural phenomena are derived, sometimes, from experimentation and observation. Mathematical methods can be applied to expand on these formulations, and develop them into better models. In the year 1856, the French hydraulic engineer Henry Darcy performed experiments, measuring water flow through a column of sand. He discovered and described a fundamental law: the linear relation between pressure difference and flow rate – known today as Darcy’s law. We describe the law and the evolution of its modern formulation. We furthermore sketch some current mathematical research related to Darcy’s law.
- ItemLimits of graph sequences(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Klimošová, TerezaGraphs are simple mathematical structures used to model a wide variety of real-life objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to efficiently represent and study properties of extremely large graphs led to the development of the theory of graph limits.
- ItemMixed volumes and mixed integrals(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Rotem, LiranIn recent years, mathematicians have developed new approaches to study convex sets: instead of considering convex sets themselves, they explore certain functions or measures that are related to them. Problems from convex geometry become thereby accessible to analytic and probabilistic tools, and we can use these tools to make progress on very difficult open problems. We discuss in this Snapshot such a functional extension of some “volumes” which measure how “big” a set is. We recall the construction of “intrinsic volumes”, discuss the fundamental inequalities between them, and explain the functional extensions of these results.
- ItemMinimizing energy(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Breiner, ChristineWhat is the most efficient way to fence land when you’ve only got so many metres of fence? Or, to put it differently, what is the largest area bounded by a simple closed planar curve of fixed length? We consider the answer to this question and others like it, making note of recent results in the same spirit.
- ItemA short story on optimal transport and its many applications(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Santambrogio, FilippoWe present some examples of optimal transport problems and of applications to different sciences (logistics, economics, image processing, and a little bit of evolution equations) through the crazy story of an industrial dynasty regularly asking advice from an exotic mathematician.
- ItemThe mathematics of aquatic locomotion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Tucsnak, MariusAquatic locomotion is a self-propelled motion through a liquid medium. It can be of biological nature (fish, microorganisms,. . .) or performed by robotic swimmers. This snapshot aims to introduce the reader to some of the challenges raised by the mathematical modelling of aquatic locomotion, even in seemingly very simple cases.