• From computer algorithms to quantum field theory: an introduction to operads 

      [SNAP-2015-017-EN] Krähmer, Ulrich (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 ...
    • Modelling the spread of brain tumours 

      [SNAP-2015-013-EN] Swan, Amanda; Murtha, Albert (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      The study of mathematical biology attempts to use mathematical models to draw useful conclusions about biological systems. Here, we consider the modelling of brain tumour spread with the ultimate goal of improving treatment ...
    • Domino tilings of the Aztec diamond 

      [SNAP-2015-016-EN] Rué, Juanjo (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 ...
    • The mystery of sleeping sickness – why does it keep waking up? 

      [SNAP-2015-015-EN] Funk, Sebastian (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      Sleeping sickness is a neglected tropical disease that affects rural populations in Africa. Deadly when untreated, it is being targeted for elimination through case finding and treatment. Yet, fundamental questions about ...
    • Chaos and chaotic fluid mixing 

      [SNAP-2015-005-EN] Solomon, Tom (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      Very simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with behavior that is sensitive to small deviations in the initial conditions. We illustrate this with a single recurrence ...
    • Ideas of Newton-Okounkov bodies 

      [SNAP-2015-008-EN] Kiritchenko, Valentina; Timorin, Vladlen; Smirnov, Evgeny (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will ...
    • Billiards and flat surfaces 

      [SNAP-2015-001-ENSNAP-2015-001-DE] Davis, Diana (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      [also available in German] Billiards, the study of a ball bouncing around on a table, is a rich area of current mathematical research. We discuss questions and results on billiards, and on the related topic of flat surfaces.
    • Fokus-Erkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse 

      [SNAP-2016-008-DE] Deistler, Manfred; Graef, Andreas (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 ...
    • High performance computing on smartphones 

      [SNAP-2016-006-EN] Patera, Anthony T.; Urban, Karsten (Mathematisches Forschungsinstitut Oberwolfach, 2016)
      Nowadays there is a strong demand to simulate even real-world 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 ...
    • Das Problem der Kugelpackung 

      [SNAP-2016-004-DE] Dostert, Maria; Krupp, Stefan; Rolfes, Jan Hendrik (Mathematisches Forschungsinstitut Oberwolfach, 2016)
      Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3-dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen ...
    • Symmetry and characters of finite groups 

      [SNAP-2016-005-EN] Giannelli, Eugenio; Taylor, Jay (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 ...
    • Random sampling of domino and lozenge tilings 

      [SNAP-2016-002-EN] Fusy, Éric (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 ...
    • On the containment problem 

      [SNAP-2016-003-EN] Szemberg, Tomasz; Szpond, Justyna (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 ...
    • Footballs and donuts in four dimensions 

      [SNAP-2016-012-EN] Klee, Steven (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 ...
    • Prime tuples in function fields 

      [SNAP-2016-010-EN] Bary-Soroker, Lior (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 ...
    • The Willmore Conjecture 

      [SNAP-2016-011-EN] Nowaczyk, Nikolai (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 donut-shaped surface and how the intuitive notion of bending has been studied by mathematics over time.
    • Swarming robots 

      [SNAP-2016-001-EN] Egerstedt, Magnus (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 ...
    • The adaptive finite element method 

      [SNAP-2016-013-EN] Gallistl, Dietmar (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 ...
    • Profinite groups 

      [SNAP-2016-014-EN] Bartholdi, Laurent (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, ...
    • Towards a Mathematical Theory of Turbulence in Fluids 

      [SNAP-2016-015-EN] Bedrossian, Jacob (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 ...