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We describe some of the beautiful mathematical structures that arise from the study of the associativity equation. Our journey takes us from combinatorics to abstract algebra, with brief excursions through geometry and topology along the way.
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Richard Brauer (1901–1977) was a German–American mathematician who is regarded as the founder of a highly active mathematical area known as modular representation theory. This area grew from group theory, which can be thought of as the mathematical study of symmetries. In this snapshot, we hope to impress on the reader the legacy left by Brauer and celebrate the 60th anniversary of “Brauer’s problems”, a list of 43 conjectures and objectives suggested by Brauer in 1963. These problems inspired an entire branch within character theory, studying “local-global conjectures”.
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This snapshot introduces the theory of trisections of smooth 4-manifolds, an area of exploration in low-dimensional topology aiming to make four-dimensional spaces more understandable. Along the way, we discuss the concepts of topology, dimension, manifolds, and more!
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Convex polytopes are geometric objects that look deceptively simple. They occur everywhere in mathematics and have practical applications in everyday life – like organizing your grocery shopping list. In this snapshot, you get into contact with a long-standing, unsolved question in mathematics, which you can explore interactively.
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In the ’70s, physicists introduced a new type of symmetry – supersymmetry – to address some unresolved issues in particle physics models. Its mathematical foundations involve the representation theory of the associated symmetry groups, called supergroups.
Our aim is to understand fusion rules, which describe how a combination of two physical systems can be broken down into more fundamental building blocks. Although the answer is largely unknown, we can get approximate answers in some cases.
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Uncertainty – as opposed to risk – is used to describe events to which we are not able to assign a probability due to lack of information. Instead of assigning a probability to an uncertain event, we only assume that such an event is possible or that its probability is within some range. We illustrate the effects of the inclusion of uncertainty in modeling by looking at simple cases of an optimal investment problem.
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This snapshot compares two techniques of shuffling a deck of cards, asking how long it will take to shuffle the cards until a “well-mixed deck” is obtained. Surprisingly, the number of shuffles can be very different for very similar looking shuffling techniques.
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Deciding which mall, hospital or school is closest to us is a problem we face everyday. It even comes on holidays with us, when we optimize our plans to make sure that we have enough time to visit all the attractions we want to see. In this article, we show how concepts from metric algebraic geometry help us to rise to this task while planning a weekend trip to the Black Forest.
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Differential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic effects into account. Surprisingly, despite the presence of randomness, our probabilistic prediction of future states is stable with respect to changes in the surrounding environment, even if the original prediction was unstable. This snapshot will unveil the core mathematical mechanism underlying this “regularisation by noise” phenomenon.
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The wave equation in Euclidean spaces describes many natural phenomena such as sound, light, or water waves. We explore how its solutions are related to the geometric problem of how long thin cylinders can intersect each other and discuss a related open problem.