• 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.
    • Billiards and flat surfaces 

      [SNAP-2015-001-ENSNAP-2015-001-DE] Davis, Diana (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      [also available in German] Billiards, the study of a ball bouncing around on a table, is a rich area of current mathematical research. We discuss questions and results on billiards, and on the related topic of flat surfaces.
    • 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 ...
    • The codimension 

      [SNAP-2018-009-EN] Lerario, Antonio (Mathematisches Forschungsinstitut Oberwolfach, 2018-06-19)
      In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one.
    • Configuration spaces and braid groups 

      [SNAP-2019-011-EN] Jiménez Rolland, Rita; Xicoténcatl, Miguel A. (Mathematisches Forschungsinstitut Oberwolfach, 2019-10-08)
      In this snapshot we introduce configuration spaces and explain how a mathematician studies their ‘shape’. This will lead us to consider paths of configurations and braid groups, and to explore how algebraic properties of ...
    • Describing distance: from the plane to spectral triples 

      [SNAP-2021-009-EN] Arici, Francesca; Mesland, Bram (Mathematisches Forschungsinstitut Oberwolfach, 2021-12-31)
      Geometry draws its power from the abstract structures that govern the shapes found in the real world. These abstractions often provide deeper insights into the underlying mathematical objects. In this snapshot, we give a ...
    • Estimating the volume of a convex body 

      [SNAP-2018-015-EN] Baldin, Nicolai (Mathematisches Forschungsinstitut Oberwolfach, 2018-12-30)
      Sometimes the volume of a convex body needs to be estimated, if we cannot calculate it analytically. We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape.
    • 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 ...
    • Footballs and donuts in four dimensions 

      [SNAP-2016-012-EN] Klee, Steven (Mathematisches Forschungsinstitut Oberwolfach, 2016)
      In this snapshot, we explore connections between the mathematical areas of counting and geometry by studying objects called simplicial complexes. We begin by exploring many familiar objects in our three dimensional world ...
    • From Betti numbers to ℓ²-Betti numbers 

      [SNAP-2020-001-EN] Kammeyer, Holger; Sauer, Roman (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-15)
      We provide a leisurely introduction to ℓ²-Betti numbers, which are topological invariants, by relating them to their much older cousins, Betti numbers. In the end we present an open research problem about ℓ²-Betti numbers.
    • 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 ...
    • 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 ...
    • Higgs bundles without geometry 

      [SNAP-2020-008-EN] Rayan, Steven; Schaposnik, Laura P. (Mathematisches Forschungsinstitut Oberwolfach, 2020-09-29)
      Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal ...
    • The Interaction of Curvature and Topology 

      [SNAP-2019-020-EN] Kordaß, Jan-Bernhard (Mathematisches Forschungsinstitut Oberwolfach, 2019-12-18)
      In this snapshot we will outline the mathematical notion of curvature by means of comparison geometry. We will then try to address questions as the ways in which curvature might influence the topology of a space, and ...
    • 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 ...
    • Lagrangian mean curvature flow 

      [SNAP-2021-011-EN] Lotay, Jason D. (Mathematisches Forschungsinstitut Oberwolfach, 2021-09-16)
      Lagrangian mean curvature flow is a powerful tool in modern mathematics with connections to topics in analysis, geometry, topology and mathematical physics. I will describe some of the key aspects of Lagrangian mean curvature ...