• 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 ...
    • 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 ...
    • Computing the long term evolution of the solar system with geometric numerical integrators 

      [SNAP-2017-009-EN] Fiorelli Vilmart, Shaula; Vilmart, Gilles (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-27)
      Simulating the dynamics of the Sun–Earth–Moon system with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the ...
    • 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 ...
    • Counting self-avoiding walks on the hexagonal lattice 

      [SNAP-2019-006-EN] Duminil-Copin, Hugo (Mathematisches Forschungsinstitut Oberwolfach, 2019-06-04)
      In how many ways can you go for a walk along a lattice grid in such a way that you never meet your own trail? In this snapshot, we describe some combinatorial and statistical aspects of these so-called self-avoiding ...
    • 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. ...
    • How to choose a winner: the mathematics of social choice 

      [SNAP-2015-009-ENSNAP-2015-009-DE] Powers, Victoria Ann (Mathematisches Forschungsinstitut Oberwolfach, 2015)
      [also available in German] Suppose a group of individuals wish to choose among several options, for example electing one of several candidates to a political office or choosing the best contestant in a skating competition. ...
    • Limits of graph sequences 

      [SNAP-2019-010-EN] Klimošová, Tereza (Mathematisches Forschungsinstitut Oberwolfach, 2019-09-04)
      Graphs are simple mathematical structures used to model a wide variety of real-life objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to efficiently represent ...
    • 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 ...
    • Nonlinear Acoustics 

      [SNAP-2019-008-EN] Kaltenbacher, Barbara; Brunnhuber, Rainer (Mathematisches Forschungsinstitut Oberwolfach, 2019-09-04)
      Nonlinear acoustics has been a topic of research for more than 250 years. Driven by a wide range and a large number of highly relevant industrial and medical applications, this area has expanded enormously in the last ...
    • 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 ...
    • Positive Scalar Curvature and Applications 

      [SNAP-2019-004-EN] Rosenberg, Jonathan; Wraith, David (Mathematisches Forschungsinstitut Oberwolfach, 2019-04-25)
      We introduce the idea of curvature, including how it developed historically, and focus on the scalar curvature of a manifold. A major current research topic involves understanding positive scalar curvature. We discuss ...
    • Random permutations 

      [SNAP-2019-007-EN] Betz, Volker (Mathematisches Forschungsinstitut Oberwolfach, 2019-07-12)
      100 people leave their hats at the door at a party and pick up a completely random hat when they leave. How likely is it that at least one of them will get back their own hat? If the hats carry name tags, how difficult ...
    • Random sampling of domino and lozenge tilings 

      [SNAP-2016-002-EN] Fusy, Éric (Mathematisches Forschungsinstitut Oberwolfach, 2016)
      A grid region is (roughly speaking) a collection of “elementary cells” (squares, for example, or triangles) in the plane. One can “tile” these grid regions by arranging the cells in pairs. In this snapshot we review different ...
    • 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 ...
    • Spaces of Riemannian metrics 

      [SNAP-2017-010-EN] Bustamante, Mauricio; Kordaß, Jan-Bernhard (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-28)
      Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the ...
    • Topological recursion 

      [SNAP-2018-002-EN] Sułkowski, Piotr (Mathematisches Forschungsinstitut Oberwolfach, 2018-03-05)
      In this snapshot we present the concept of topological recursion – a new, surprisingly powerful formalism at the border of mathematics and physics, which has been actively developed within the last decade. After introducing ...
    • Winkeltreue zahlt sich aus 

      [SNAP-2017-001-DE] Günther, Felix (Mathematisches Forschungsinstitut Oberwolfach, 2017-08-23)
      Nicht nur Seefahrerinnen, auch Computergrafikerinnen und Physikerinnen wissen Winkeltreue zu schätzen. Doch beschränkte Rechenkapazitäten und Vereinfachungen in theoretischen Modellen erfordern es, winkeltreue Abbildungen ...