• Billard und ebene Flächen 

      [SNAP-2015-001-DE] Davis, Diana (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      Billard, die Zick-Zack-Bewegungen eines Balls auf einem Tisch, ist ein reichhaltiges Feld gegenwärtiger mathematischer Forschung. In diesem Artikel diskutieren wir Fragen und Antworten zum Thema Billard, und zu dem damit ...
    • Billiards and flat surfaces 

      [SNAP-2015-001-EN] Davis, Diana (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      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.
    • Biological shape analysis with geometric statistics and learning 

      [SNAP-2022-008-EN] Utpala, Saiteja; Miolane, Nina (Mathematisches Forschungsinstitut Oberwolfach, 2022-10-25)
      The 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 ...
    • Braid groups, the Yang–Baxter equation, and subfactors 

      [SNAP-2021-005-EN] Lechner, Gandalf (Mathematisches Forschungsinstitut Oberwolfach, 2021)
      The Yang–Baxter equation is a fascinating equation that appears in many areas of physics and mathematics and is best represented diagramatically. This snapshot connects the mathematics of braiding hair to the Yang–Baxter ...
    • $C^*$-algebras: structure and classification 

      [SNAP-2021-002-EN] Kerr, David (Mathematisches Forschungsinstitut Oberwolfach, 2021-02-23)
      The theory of $C^*$-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 ...
    • 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 ...
    • Characterizations of intrinsic volumes on convex bodies and convex functions 

      [SNAP-2022-011-EN] Mussnig, Fabian (Mathematisches Forschungsinstitut Oberwolfach, 2022-12-08)
      If 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 ...
    • Charakterisierungen von inneren Volumina auf konvexen Körpern und konvexen Funktionen 

      [SNAP-2022-011-DE] Mussnig, Fabian (Mathematisches Forschungsinstitut Oberwolfach, 2023)
      Wenn wir die Größe einer zweidimensionalen Form mittels einer Zahl ausdrücken wollen, dann denken wir gewöhnlich an ihren Flächeninhalt oder ihren Umfang. Aber was macht diese Kennzahlen so besonders? Wir beantworten diese ...
    • Closed geodesics on surfaces 

      [SNAP-2022-013-EN] Dozier, Benjamin (Mathematisches Forschungsinstitut Oberwolfach, 2022-12-08)
      We 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 ...
    • Closed geodesics on surfaces and Riemannian manifolds 

      [SNAP-2017-005-EN] Radeschi, Marco (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-07)
      Geodesics are special paths in surfaces and so-called Riemannian manifolds which connect close points in the shortest way. Closed geodesics are geodesics which go back to where they started. In this snapshot we talk ...
    • The codimension 

      [SNAP-2018-009-EN] Lerario, Antonio (Mathematisches Forschungsinstitut Oberwolfach, 2018-06-19)
      In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one.
    • Computational Optimal Transport 

      [SNAP-2017-008-EN] Solomon, Justin (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-21)
      Optimal transport is the mathematical discipline of matching supply to demand while minimizing shipping costs. This matching problem becomes extremely challenging as the quantity of supply and demand points increases; ...
    • Computing the long term evolution of the solar system with geometric numerical integrators 

      [SNAP-2017-009-EN] Fiorelli Vilmart, Shaula; Vilmart, Gilles (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-27)
      Simulating the dynamics of the Sun–Earth–Moon system with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the ...
    • Computing with symmetries 

      [SNAP-2018-003-EN] Roney-Dougal, Colva M. (Mathematisches Forschungsinstitut Oberwolfach, 2018-03-06)
      Group theory is the study of symmetry, and has many applications both within and outside mathematics. In this snapshot, we give a brief introduction to symmetries, and how to compute with them.
    • Configuration spaces and braid groups 

      [SNAP-2019-011-EN] Jiménez Rolland, Rita; Xicoténcatl, Miguel A. (Mathematisches Forschungsinstitut Oberwolfach, 2019-10-08)
      In this snapshot we introduce configuration spaces and explain how a mathematician studies their ‘shape’. This will lead us to consider paths of configurations and braid groups, and to explore how algebraic properties of ...
    • Counting self-avoiding walks on the hexagonal lattice 

      [SNAP-2019-006-EN] Duminil-Copin, Hugo (Mathematisches Forschungsinstitut Oberwolfach, 2019-06-04)
      In how many ways can you go for a walk along a lattice grid in such a way that you never meet your own trail? In this snapshot, we describe some combinatorial and statistical aspects of these so-called self-avoiding ...
    • Curriculum development in university mathematics: where mathematicians and education collide 

      [SNAP-2015-011-EN] Sangwin, Christopher J. (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      This snapshot looks at educational aspects of the design of curricula in mathematics. In particular, we examine choices textbook authors have made when introducing the concept of the completness of the real numbers. Can ...
    • Curvatura escalar positiva y aplicaciones 

      [SNAP-2019-004-ES] Rosenberg, Jonathan; Wraith, David (Mathematisches Forschungsinstitut Oberwolfach, 2021)
      Introducimos la idea de curvatura, incluyendo su desarrollo histórico, y nos enfocamos en la curvatura escalar de una variedad. Uno de los temas principales de investigación actual es entender la curvatura escalar positiva. ...
    • Cutoff Phenomenon: Surprising Behaviour in Card Shuffling and other Markov Chains 

      [SNAP-2023-005-EN] Baraquin, Isabelle; Lafrenière, Nadia; Schuh, Katharina (Mathematisches Forschungsinstitut Oberwolfach, 2023-12-21)
      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 ...
    • Darcy's law and groundwater flow modelling 

      [SNAP-2015-007-EN] Schweizer, Ben (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      Formulations 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, ...