Browsing by Snapshot Mathematical Subject "Geometry and Topology"

Now showing items 21-40 of 63

    • 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 ...

    • Fibrés de Higgs sans géométrie 

      [SNAP-2020-008-FR] Rayan, Steven; Schaposnik, Laura P. (Mathematisches Forschungsinstitut Oberwolfach, 2024-03-05)
      Les fibrés de Higgs sont apparus il y a quelques décennies comme solutions de certaines équations en physique, et ils ont attiré beaucoup d’attention en géométrie comme dans d’autres domaines des mathématiques et de la ...

    • 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 ...

    • The five Platonic solids and their connection to root systems 

      [SNAP-2025-004-EN] Böhm, Sören (Mathematisches Forschungsinstitut Oberwolfach, 2025-09-05)
      Platonic solids have fascinated humans for thousands of years. In ancient times, they were associated with the elements fire, air, water, earth, and aether. These solids are completely symmetrical three-dimensional polyhedra. ...

    • Five ways to spell ADE 

      [SNAP-2025-003-EN] Kaufman, Dani (Mathematisches Forschungsinstitut Oberwolfach, 2025-07-30)
      The solutions to a surprising number of mathematical questions can be classified by the ADE Coxeter–Dynkin diagrams. This snapshot will show you a selection of these questions and how they correspond to the ADE Coxeter–Dynkin ...

    • 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 ...

    • 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 ...

    • 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 stroll ...

    • How big is my slice of cheese? 

      [SNAP-2026-006-EN] Meroni, Chiara (Mathematisches Forschungsinstitut Oberwolfach, 2026-05-19)
      In this snapshot, we introduce the study of slices of polytopes – geometric shapes with flat sides – and examine the area of these slices. This is connected to combinatorics and polynomials and is surprisingly complex, ...

    • 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 ...

    • Is there a smooth lattice polytope which does not have the integer decomposition property? 

      [SNAP-2025-008-EN] Hofscheier, Johannes; Kasprzyk, Alexander (Mathematisches Forschungsinstitut Oberwolfach, 2025-12-16)
      We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the ...

    • 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 ...

    • Matrixfaktorisierungen 

      [SNAP-2014-002-DE] Lerche, Wolfgang (Mathematisches Forschungsinstitut Oberwolfach, 2014)
      Im Folgenden soll ein kurzer Abriss des Themas Matrixfaktorisierungen gegeben werden. Wir werden darlegen, warum dieses recht simple Konzept zu erstaunlich tiefen mathematischen Gedankengängen führt und auch in der modernen ...

    • Minimizing energy 

      [SNAP-2015-002-EN] Breiner, Christine (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      What is the most efficient way to fence land when you’ve only got so many metres of fence? Or, to put it differently, what is the largest area bounded by a simple closed planar curve of fixed length? We consider the answer ...

    • Mixed volumes and mixed integrals 

      [SNAP-2018-014-EN] Rotem, Liran (Mathematisches Forschungsinstitut Oberwolfach, 2018-12-29)
      In recent years, mathematicians have developed new approaches to study convex sets: instead of considering convex sets themselves, they explore certain functions or measures that are related to them. Problems from ...