• Snake graphs, perfect matchings and continued fractions 

      [SNAP-2019-001-EN] Schiffler, Ralf (Mathematisches Forschungsinstitut Oberwolfach, 2019-02-13)
      A continued fraction is a way of representing a real number by a sequence of integers. We present a new way to think about these continued fractions using snake graphs, which are sequences of squares in the plane. You ...
    • On radial basis functions 

      [SNAP-2019-002-EN] Buhmann, Martin; Jäger, Janin (Mathematisches Forschungsinstitut Oberwolfach, 2019-03-13)
      Many sciences and other areas of research and applications from engineering to economics require the approximation of functions that depend on many variables. This can be for a variety of reasons. Sometimes we have a ...
    • 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. ...
    • 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 ...
    • 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 ...
    • 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 ...
    • 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 ...
    • 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 ...
    • 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 ...
    • 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 ...
    • Touching the transcendentals: tractional motion from the bir th of calculus to future perspectives 

      [SNAP-2019-013-EN] Milici, Pietro (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
      When the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century ...
    • Analogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964) 

      [SNAP-2019-012-EN] Petitgirard, Loïc (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
      Throughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers ...
    • Deep Learning and Inverse Problems 

      [SNAP-2019-015-EN] Arridge, Simon; de Hoop, Maarten; Maass, Peter; Öktem, Ozan; Schönlieb, Carola; Unser, Michael (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
      Big data and deep learning are modern buzz words which presently infiltrate all fields of science and technology. These new concepts are impressive in terms of the stunning results they achieve for a large variety of ...
    • Mixed-dimensional models for real-world applications 

      [SNAP-2019-014-EN] Nordbotten, Jan Martin (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
      We explore mathematical models for physical problems in which it is necessary to simultaneously consider equations in different dimensions; these are called mixed-dimensional models. We first give several examples, and ...
    • 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 ...
    • Formation Control and Rigidity Theory 

      [SNAP-2019-017-EN] Zelazo, Daniel; Zhao, Shiyu (Mathematisches Forschungsinstitut Oberwolfach, 2019-12-11)
      Formation control is one of the fundamental coordination tasks for teams of autonomous vehicles. Autonomous formations are used in applications ranging from search-and-rescue operations to deep space exploration, with ...
    • A surprising connection between quantum mechanics and shallow water waves 

      [SNAP-2019-018-EN] Fillman, Jake; VandenBoom, Tom (Mathematisches Forschungsinstitut Oberwolfach, 2019-12-11)
      We describe a connection between quantum mechanics and nonlinear wave equations and highlight a few problems at the forefront of modern research in the intersection of these areas.
    • 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 it possible to predict the far future before the near future is known accurately? 

      [SNAP-2019-021-EN] Gander, Martin J. (Mathematisches Forschungsinstitut Oberwolfach, 2019-12-18)
      It has always been the dream of mankind to predict the future. If the future is governed by laws of physics, like in the case of the weather, one can try to make a model, solve the associated equations, and thus predict ...