Snapshots of Modern Mathematics from Oberwolfach
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Item type: Item , The Five Platonic Solids and their Connection to Root Systems(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Böhm, SörenPlatonic solids have fascinated humans for thousands of years. In ancient times, they were associated with the elements fire, air, water, earth, and aether. These solids are completely symmetrical three-dimensional polyhedra. In this snapshot, it is first explained that there can only be five such polyhedra in the three-dimensional space. For this purpose, so-called Schläfli symbols and Coxeter graphs are introduced. More precisely, the (linear) Coxeter graphs correspond to the (linear) Schläfli symbols that, in turn, correspond exactly to the regular convex polyhedra. Through this one-to-one relationship, it is possible to classify the regular convex polytopes in any dimension by exploiting the classification of Coxeter graphs.Item type: Item , Convex Polytopes and Linear Programs(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Joswig, MichaelConvex 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.Item type: Item , Trisections of Four-Dimensional Spaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Blackwell, SarahThis 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!Item type: Item , Five Ways to Spell ADE(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Kaufman, DaniThe solutions to a surprising number of mathematical questions can be classified by the ADE Coxeter–Dynkin diagrams. This snapshot will show you a selection of these questions and how they correspond to the ADE Coxeter–Dynkin diagrams.Item type: Item , Truncated Fusion Rules for Supergroups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Heidersdorf, ThorstenIn 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.Item type: Item , Brackets, Trees and the Borromean Rings(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Jasso, GustavoWe 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.Item type: Item , Brauer's Problems: 60 Years of Legacy(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Rizo, Noelia; Schaeffer Fry, Mandi A.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".Item type: Item , Why Oscillation Counts: Diophantine Approximation, Geometry, and the Fourier Transform(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Srivastava, RajulaIs it possible to approximate arbitrary points in space by vectors with rational coordinates, with which we, and computers, feel much more comfortable? If yes, can we approximate those points arbitrarily close? In this snapshot, we explore how the geometric configuration of these points influences the answers to these questions. Further, we delve into the closely related problem of counting rational vectors near surfaces. The unlikely tool which helps us in this endeavour is Fourier analysis – the study of waves and oscillations!Item type: Item , Is there a Smooth Lattice Polytope which does not have the Integer Decomposition Property?(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2025) Hofscheier, Johannes; Kasprzyk, AlexanderWe introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the two-dimensional case – including a proof of Pick's Theorem, which elegantly relates the area of a lattice polygon to the number of lattice points it contains in its interior and on its boundary.Item type: Item , Cutoff-Phänomen: überraschendes Verhalten beim Kartenmischen und bei weiteren Markovketten(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2024) Baraquin, Isabelle; Lafrenière, Nadia; Schuh, KatharinaDieser Schnappschuss vergleicht zwei Arten des Kartenmischens und untersucht, wie lange es dauert einen "gut gemischten" Kartenstapel zu erhalten. Überraschenderweise kann das Mischverhalten auch für sehr ähnlich ausschauende Kartenmischtechniken sehr unterschiedlich sein.Item type: Item , Uncertainty as an Ingredient in Financial Modeling(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2024) Korn, RalfUncertainty - 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.Item type: Item , Closed geodesics on surfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Dozier, BenjaminWe consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line paths, known as geodesics, in these geometries. We discuss the issue of counting closed geodesics; this is particularly rich for hyperbolic (negatively curved) surfaces.Item type: Item , Patterns and Waves in Theory, Experiment, and Application(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Bramburger, Jason J.In this snapshot of modern mathematics we describe some of the most prevalent waves and patterns that can arise in mathematical models and which are used to describe a number of biological, chemical, physical, and social processes. We begin by focussing on two types of patterns that do not change in time: space-filling patterns and localized patterns. We then discuss two types of waves that evolve predictably as time goes on: spreading waves and rotating waves. All our examples are motivated with real-world applications and we highlight some of the main lines of research that mathematicians pursue to better understand them.Item type: Item , What is pattern?(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Baake, Michael; Grimm, Uwe; Moody, Robert V.Pattern is ubiquitous and seems totally familiar. Yet if we ask what it is, we find a bewildering collection of answers. Here we suggest that there is a common thread, and it revolves around dynamics.Item type: Item , Characterizations of intrinsic volumes on convex bodies and convex functions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Mussnig, FabianIf we want to express the size of a two-dimensional shape with a number, then we usually think about its area or circumference. But what makes these quantities so special? We give an answer to this question in terms of classical mathematical results. We also take a look at applications and new generalizations to the setting of functions.Item type: Item , A tale of three curves(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Balakrishnan, Jennifer S.In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.Item type: Item , Route planning for bacteria(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Hellmuth, Kathrin; Klingenberg, ChristianBacteria have been fascinating biologists since their discovery in the late 17th century. By analysing their movements, mathematical models have been developed as a tool to understand their behaviour. However, adapting these models to real situations can be challenging, because the model coefficients cannot be observed directly. In this snapshot, we study this question mathematically and explain how the idea of “route planning” can be used to determine these model coefficients.Item type: Item , Randomness is Natural - an Introduction to Regularisation by Noise(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2024) Djurdjevac, Ana; Elad Altman, Henri; Rosati, TommasoDifferential 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.Item type: Item , Representations and degenerations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Dumanski, Ilya; Kiritchenko, ValentinaIn this snapshot, we explain two important mathematical concepts (representation and degeneration) in elementary terms. We will focus on the simplest meaningful examples, and motivate both concepts by study of symmetry.Item type: Item , Biological shape analysis with geometric statistics and learning(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Utpala, Saiteja; Miolane, NinaThe advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold the key to unlocking outstanding mysteries in biomedicine. This snapshot introduces the mathematical framework of geometric statistics and learning and its applications to biomedicine.