• 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 ...
    • 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.
    • 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. ...
    • Expander graphs and where to find them 

      [SNAP-2019-016-EN] Khukhro, Ana (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-22)
      Graphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical ...
    • 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 ...
    • 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 ...
    • 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 ...
    • From the dollar game to the Riemann-Roch Theorem 

      [SNAP-2021-001-EN] Lamboglia, Sara; Ulirsch, Martin (Mathematisches Forschungsinstitut Oberwolfach, 2021-02-23)
      What is the dollar game? What can you do to win it? Can you always win it? In this snapshot you will find answers to these questions as well as several of the mathematical surprises that lurk in the background, including ...
    • 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 ...
    • Invitation to quiver representation and Catalan combinatorics 

      [SNAP-2021-004-EN] Rognerud, Baptiste (Mathematisches Forschungsinstitut Oberwolfach, 2021-04-08)
      Representation theory is an area of mathematics that deals with abstract algebraic structures and has numerous applications across disciplines. In this snapshot, we will talk about the representation theory of a class ...
    • 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 Logic, Choices and Games 

      [SNAP-2019-009-EN] Oliva, Paulo (Mathematisches Forschungsinstitut Oberwolfach, 2019-09-04)
      Can we always mathematically formalise our taste and preferences? We discuss how this has been done historically in the field of game theory, and how recent ideas from logic and computer science have brought an interesting ...
    • 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 ...