• 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 ...
    • 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.
    • 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.
    • A few shades of interpolation 

      [SNAP-2017-007-EN] Szpond, Justyna (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-07)
      The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points ...
    • Friezes and tilings 

      [SNAP-2015-004-EN] Holm, Thorsten (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      Friezes have occured as architectural ornaments for many centuries. In this snapshot, we consider the mathematical analogue of friezes as introduced in the 1970s by Conway and Coxeter. Recently, infinite versions of such ...
    • From computer algorithms to quantum field theory: an introduction to operads 

      [SNAP-2015-017-EN] Krähmer, Ulrich (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      An operad is an abstract mathematical tool encoding operations on specific mathematical structures. It finds applications in many areas of mathematics and related fields. This snapshot explains the concept of an operad and ...
    • Ideas of Newton-Okounkov bodies 

      [SNAP-2015-008-EN] Kiritchenko, Valentina; Timorin, Vladlen; Smirnov, Evgeny (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will ...
    • News on quadratic polynomials 

      [SNAP-2017-002-EN] Pottmeyer, Lukas (Mathematisches Forschungsinstitut Oberwolfach, 2017-07-18)
      Many problems in mathematics have remained unsolved because of missing links between mathematical disciplines, such as algebra, geometry, analysis, or number theory. Here we introduce a recently discovered result concerning ...
    • Number theory in quantum computing 

      [SNAP-2018-012-EN] Schönnenbeck, Sebastian (Mathematisches Forschungsinstitut Oberwolfach, 2018-08-07)
      Algorithms are mathematical procedures developed to solve a problem. When encoded on a computer, algorithms must be "translated" to a series of simple steps, each of which the computer knows how to do. This task is ...
    • 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 ...
    • 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 ...
    • 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 ...
    • 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 ...