2  Snapshots of modern mathematics from Oberwolfach
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Snapshots of modern mathematics from Oberwolfach explain mathematical problems and ideas in an accessible and understandable way. They provide exciting insights into current topics of the mathematical community for everyone who is interested in modern mathematics.
The snapshots are written by participants of the scientific program at the MFO, who volunteer to explain an important aspect of their research. A team of editors assists them in communicating complicated matters to a broad audience. The MFO publishes the snapshots for free download under a Creative Commons license here and on the IMAGINARY platform.
The Snapshot project is designed to promote the understanding and appreciation of modern mathematics and mathematical research in the general public worldwide. The targeted readership consists of mathematics teachers, science journalists, undergraduate and advanced high school students. If you are interested in writing a snapshot, please contact the organizers of the program in which you are participating.
The project started in January 2014 as part of the project "Oberwolfach meets IMAGINARY". It is funded by the Klaus Tschira Foundation and has also been supported by the Oberwolfach Foundation and the MFO.
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Aperiodic Order and Spectral Properties
(Mathematisches Forschungsinstitut Oberwolfach, 20170914)Periodic 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 ... 
Winkeltreue zahlt sich aus
(Mathematisches Forschungsinstitut Oberwolfach, 20170823)Nicht nur Seefahrerinnen, auch Computergrafikerinnen und Physikerinnen wissen Winkeltreue zu schätzen. Doch beschränkte Rechenkapazitäten und Vereinfachungen in theoretischen Modellen erfordern es, winkeltreue Abbildungen ... 
News on quadratic polynomials
(Mathematisches Forschungsinstitut Oberwolfach, 20170718)Many 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 ... 
Towards a Mathematical Theory of Turbulence in Fluids
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Fluid 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 ... 
Profinite groups
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Profinite 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, ... 
The adaptive finite element method
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Computer simulations of many physical phenomena rely on approximations by models with a finite number of unknowns. The number of these parameters determines the computational effort needed for the simulation. On the other ... 
Footballs and donuts in four dimensions
(Mathematisches Forschungsinstitut Oberwolfach, 2016)In 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 ... 
The Willmore Conjecture
(Mathematisches Forschungsinstitut Oberwolfach, 2016)The Willmore problem studies which torus has the least amount of bending energy. We explain how to think of a torus as a donutshaped surface and how the intuitive notion of bending has been studied by mathematics over time. 
Prime tuples in function fields
(Mathematisches Forschungsinstitut Oberwolfach, 2016)How many prime numbers are there? How are they distributed among other numbers? These are questions that have intrigued mathematicians since ancient times. However, many questions in this area have remained unsolved, and ... 
Polyhedra and commensurability
(Mathematisches Forschungsinstitut Oberwolfach, 2016)This 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 ... 
FokusErkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Viele 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 ... 
Wie steuert man einen Kran?
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Die 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 MasseFederSystems illustrieren wir diese ... 
High performance computing on smartphones
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Nowadays there is a strong demand to simulate even realworld 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 ... 
Symmetry and characters of finite groups
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Over 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 ... 
Das Problem der Kugelpackung
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen ... 
On the containment problem
(Mathematisches Forschungsinstitut Oberwolfach, 2016)Mathematicians 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 ... 
Random sampling of domino and lozenge tilings
(Mathematisches Forschungsinstitut Oberwolfach, 2016)A 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 ... 
Swarming robots
(Mathematisches Forschungsinstitut Oberwolfach, 2016)When lots of robots come together to form shapes, spread in an area, or move in one direction, their motion has to be planned carefully. We discuss how mathematicians devise strategies to help swarms of robots behave like ... 
From computer algorithms to quantum field theory: an introduction to operads
(Mathematisches Forschungsinstitut Oberwolfach, 2015)An operad is an abstract mathematical tool encoding operations on specific mathematical structures. It finds applications in many areas of mathematics and related fields. This snapshot explains the concept of an operad and ... 
Domino tilings of the Aztec diamond
(Mathematisches Forschungsinstitut Oberwolfach, 2015)Imagine you have a cutout from a piece of squared paper and a pile of dominoes, each of which can cover exactly two squares of the squared paper. How many different ways are there to cover the entire paper cutout with ...