• 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 ...
    • Algebra, matrices, and computers 

      [SNAP-2019-005-EN] Detinko, Alla; Flannery, Dane; Hulpke, Alexander (Mathematisches Forschungsinstitut Oberwolfach, 2019-05-03)
      What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that ...
    • The Algebraic Statistics of an Oberwolfach Workshop 

      [SNAP-2018-001-EN] Seigal, Anna (Mathematisches Forschungsinstitut Oberwolfach, 2018-02-27)
      Algebraic Statistics builds on the idea that statistical models can be understood via polynomials. Many statistical models are parameterized by polynomials in the model parameters; others are described implicitly by ...
    • Analogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964) 

      [SNAP-2019-012-EN] Petitgirard, Loïc (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
      Throughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers ...
    • Aperiodic Order and Spectral Properties 

      [SNAP-2017-003-EN] Baake, Michael; Damanik, David; Grimm, Uwe (Mathematisches Forschungsinstitut Oberwolfach, 2017-09-14)
      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 ...
    • Arrangements of lines 

      [SNAP-2014-005-EN] Harbourne, Brian; Szemberg, Tomasz (Mathematisches Forschungsinstitut Oberwolfach, 2014)
      We discuss certain open problems in the context of arrangements of lines in the plane.
    • Billiards and flat surfaces 

      [SNAP-2015-001-EN] 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.
    • 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 ...
    • 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 ...
    • 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, ...
    • Data assimilation: mathematics for merging models and data 

      [SNAP-2018-011-EN] Morzfeld, Matthias; Reich, Sebastian (Mathematisches Forschungsinstitut Oberwolfach, 2018-07-10)
      When you describe a physical process, for example, the weather on Earth, or an engineered system, such as a self-driving car, you typically have two sources of information. The first is a mathematical model, and the ...
    • Deep Learning and Inverse Problems 

      [SNAP-2019-015-EN] Arridge, Simon; de Hoop, Maarten; Maass, Peter; Öktem, Ozan; Schönlieb, Carola; Unser, Michael (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
      Big data and deep learning are modern buzz words which presently infiltrate all fields of science and technology. These new concepts are impressive in terms of the stunning results they achieve for a large variety of ...
    • Diophantine equations and why they are hard 

      [SNAP-2019-003-EN] Pasten, Hector (Mathematisches Forschungsinstitut Oberwolfach, 2019-04-24)
      Diophantine equations are polynomial equations whose solutions are required to be integer numbers. They have captured the attention of mathematicians during millennia and are at the center of much of contemporary research. ...