• 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 ...
    • 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 ...
    • Polyhedra and commensurability 

      [SNAP-2016-009-EN] Guglielmetti, Rafael; Jacquement, Matthieu (Mathematisches Forschungsinstitut Oberwolfach, 2016)
      This snapshot introduces the notion of commensurability of polyhedra. At its bottom, this concept can be developed from constructions with paper, scissors, and glue. Starting with an elementary example, we formalize it ...
    • 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 ...
    • 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 ...
    • Le problème ternaire de Goldbach 

      [SNAP-2014-003-FR] Helfgott, Harald (Mathematisches Forschungsinstitut Oberwolfach, 2024)
      Leonhard Euler (1707–1783), l’un des plus grands mathématiciens du XVIIIe siècle et de tous les temps, entretenait une correspondance régulière avec l’un de ses amis: Christian Goldbach (1690–1764), un amateur polymathe ...
    • 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, ...
    • Prony’s method: an old trick for new problems 

      [SNAP-2018-004-EN] Sauer, Tomas (Mathematisches Forschungsinstitut Oberwolfach, 2018-03-06)
      In 1795, French mathematician Gaspard de Prony invented an ingenious trick to solve a recovery problem, aiming at reconstructing functions from their values at given points, which arose from a specific application in ...
    • Quantum symmetry 

      [SNAP-2020-005-EN] Weber, Moritz (Mathematisches Forschungsinstitut Oberwolfach, 2020-06-04)
      In mathematics, symmetry is usually captured using the formalism of groups. However, the developments of the past few decades revealed the need to go beyond groups: to “quantum groups”. We explain the passage from ...
    • Random matrix theory: Dyson Brownian motion 

      [SNAP-2020-002-EN] Finocchio, Gianluca (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-15)
      The theory of random matrices was introduced by John Wishart (1898–1956) in 1928. The theory was then developed within the field of nuclear physics from 1955 by Eugene Paul Wigner (1902–1995) and later by Freeman John ...
    • 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 ...
    • 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 ...
    • Snake graphs, perfect matchings and continued fractions 

      [SNAP-2019-001-EN] Schiffler, Ralf (Mathematisches Forschungsinstitut Oberwolfach, 2019-02-13)
      A continued fraction is a way of representing a real number by a sequence of integers. We present a new way to think about these continued fractions using snake graphs, which are sequences of squares in the plane. You ...
    • Solving quadratic equations in many variables 

      [SNAP-2017-012-EN] Tignol, Jean-Pierre (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-30)
      Fields are number systems in which every linear equation has a solution, such as the set of all rational numbers $\mathbb{Q}$ or the set of all real numbers $\mathbb{R}$. All fields have the same properties in relation ...
    • Special values of zeta functions and areas of triangles 

      [SNAP-2015-010-EN] Kramer, Jürg; Pippich, Anna-Maria von (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      In this snapshot we give a glimpse of the interplay of special values of zeta functions and volumes of triangles. Special values of zeta functions and their generalizations arise in the computation of volumes of moduli ...
    • Swallowtail on the shore 

      [SNAP-2014-007-EN] Buchweitz, Ragnar-Olaf; Faber, Eleonore (Mathematisches Forschungsinstitut Oberwolfach, 2014)
      Platonic solids, Felix Klein, H.S.M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not ...
    • 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 ...
    • A tale of three curves 

      [SNAP-2022-010-EN] Balakrishnan, Jennifer S. (Mathematisches Forschungsinstitut Oberwolfach, 2022-10-27)
      In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th ...