• 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 ...
    • Algebras and Quantum Games 

      [SNAP-2023-004-EN] Paulsen, Vern I. (Mathematisches Forschungsinstitut Oberwolfach, 2023-11-28)
      Everyone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out!
    • 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 ...
    • 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 ...
    • Characterizations of intrinsic volumes on convex bodies and convex functions 

      [SNAP-2022-011-ENSNAP-2022-011-DE] 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 ...
    • 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 ...
    • 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. ...
    • Emergence in biology and social sciences 

      [SNAP-2022-001-EN] Hoffmann, Franca; Merino-Aceituno, Sara (Mathematisches Forschungsinstitut Oberwolfach, 2022-03-31)
      Mathematics is the key to linking scientific knowledge at different scales: from microscopic to macroscopic dynamics. This link gives us understanding on the emergence of observable patterns like flocking of birds, leaf ...
    • Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften 

      [SNAP-2023-003-DE] Saberi, Ingmar (Mathematisches Forschungsinstitut Oberwolfach, 2023-09-19)
      Wir werden einige grundlegende Ideen der Eichtheorie und der dazugehörigen Differentialtopologie erkunden. Damit kann sich die Leserin ein Bild des Modulraums flacher Zusammenhänge machen und ihn mit den physikalisch ...
    • Finite geometries: pure mathematics close to applications 

      [SNAP-2021-010-EN] Storme, Leo (Mathematisches Forschungsinstitut Oberwolfach, 2021-09-22)
      The research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these ...
    • Geometry behind one of the Painlevé III differential equations 

      [SNAP-2018-010-EN] Hertling, Claus (Mathematisches Forschungsinstitut Oberwolfach, 2018-06-20)
      The Painlevé equations are second order differential equations, which were first studied more than 100 years ago. Nowadays they arise in many areas in mathematics and mathematical physics. This snapshot discusses the ...
    • Jewellery from tessellations of hyperbolic space 

      [SNAP-2022-005-EN] Gangl, Herbert (Mathematisches Forschungsinstitut Oberwolfach, 2022-06-02)
      In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional ...
    • Patterns and Waves in Theory, Experiment, and Application 

      [SNAP-2023-001-EN] Bramburger, Jason J. (Mathematisches Forschungsinstitut Oberwolfach, 2023-07-04)
      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 ...
    • The Periodic Tables of Algebraic Geometry 

      [SNAP-2023-002-EN] Belmans, Pieter (Mathematisches Forschungsinstitut Oberwolfach, 2023-09-04)
      To understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use ...
    • Reflections on hyperbolic space 

      [SNAP-2021-007-EN] Haensch, Anna (Mathematisches Forschungsinstitut Oberwolfach, 2021-08-24)
      In school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', ...
    • Representations and degenerations 

      [SNAP-2022-007-EN] Dumanski, Ilya; Kiritchenko, Valentina (Mathematisches Forschungsinstitut Oberwolfach, 2022-10-25)
      In 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.
    • The Robinson–Schensted algorithm 

      [SNAP-2022-002-EN] Thomas, Hugh (Mathematisches Forschungsinstitut Oberwolfach, 2022-05-06)
      I am going to describe the Robinson–Schensted algorithm which transforms a permutation of the numbers from 1 to n into a pair of combinatorial objects called “standard Young tableaux”. I will then say a little bit about a ...
    • Route planning for bacteria 

      [SNAP-2022-012-EN] Hellmuth, Kathrin; Klingenberg, Christian (Mathematisches Forschungsinstitut Oberwolfach, 2022-12-08)
      Bacteria 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 ...
    • Searching for the monster in the trees 

      [SNAP-2022-003-EN] Craven, David A. (Mathematisches Forschungsinstitut Oberwolfach, 2022-04-13)
      The Monster finite simple group is almost unimaginably large, with about 8 × 1053 elements in it. Trying to understand such an immense object requires both theory and computer programs. In this snapshot, we discuss finite ...