• 4 = 2 × 2, or the Power of Even Integers in Fourier Analysis 

      [SNAP-2023-006-EN] Negro, Giuseppe; Oliveira e Silva, Diogo (Mathematisches Forschungsinstitut Oberwolfach, 2023-12-30)
      We describe how simple observations related to vectors of length 1 recently led to the proof of an important mathematical fact: the sharp Stein–Tomas inequality from Fourier restriction theory, a pillar of modern harmonic ...
    • 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-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 ...
    • 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 ...
    • 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 ...
    • Espacios de métricas Riemannianas 

      [SNAP-2017-010-ES] Bustamante, Mauricio; Kordaß, Jan-Bernhard (Mathematisches Forschungsinstitut Oberwolfach, 2021)
      Las métricas riemannianas dan a las variedades suaves, como las superficies, propiedades geométricas intrínsecas, por ejemplo la curvatura. También permiten medir cantidades como distancias, ángulos y volúmenes. Estas son ...
    • 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 ...
    • The Geometry of Fair Division 

      [SNAP-2023-007-EN] Frick, Florian (Mathematisches Forschungsinstitut Oberwolfach, 2023-12-30)
      How can we fairly divide a necklace with various types of beads? We use this problem as a motivating example to explain how geometry naturally appears in solutions of non-geometric problems. The strategy we develop to solve ...
    • Geproci Sets: a New Perspective in Algebraic Geometry 

      [SNAP-2023-008-EN] Chiantini, Luca; Harbourne, Brian (Mathematisches Forschungsinstitut Oberwolfach, 2023-12-30)
      Geproci sets arise from applying the perspective of inverse scattering problems to algebraic geometry. Analogous to the reconstruction of an object from multiple X-ray images, we aim at a classification of sets with certain ...
    • 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 ...