Browsing 2  Snapshots of modern mathematics from Oberwolfach by Title
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Geometry behind one of the Painlevé III differential equations
[SNAP2018010EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180620)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
[SNAP2020008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200929)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 ... 
High performance computing on smartphones
[SNAP2016006EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Nowadays there is a strong demand to simulate even realworld engineering problems on small computing devices with very limited capacity, such as a smartphone. We explain, using a concrete example, how we can obtain a ... 
How to choose a winner: the mathematics of social choice
[SNAP2015009ENSNAP2015009DE] (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. ... 
Ideas of NewtonOkounkov bodies
[SNAP2015008EN] (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 NewtonOkounkov bodies of which we will ... 
The Interaction of Curvature and Topology
[SNAP2019020EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191218)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 ... 
Invitation to quiver representation and Catalan combinatorics
[SNAP2021004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210408)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 ... 
Is it possible to predict the far future before the near future is known accurately?
[SNAP2019021EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191218)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 ... 
Jewellery from tessellations of hyperbolic space
[SNAP2022005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220602)In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a numbertheoretic origin give rise to a large variety of interesting tessellations of 3dimensional ... 
The KadisonSinger problem
[SNAP2014008EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)In quantum mechanics, unlike in classical mechanics, one cannot make precise predictions about how a system will behave. Instead, one is concerned with mere probabilities. Consequently, it is a very important task to ... 
Lagrangian mean curvature flow
[SNAP2021011EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210916)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 ... 
Limits of graph sequences
[SNAP2019010EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190904)Graphs are simple mathematical structures used to model a wide variety of reallife objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to efficiently represent ... 
The mathematics of aquatic locomotion
[SNAP2018008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180619)Aquatic locomotion is a selfpropelled motion through a liquid medium. It can be of biological nature (fish, microorganisms,. . .) or performed by robotic swimmers. This snapshot aims to introduce the reader to some ... 
The Mathematics of Fluids and Solids
[SNAP2019019EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191218)Fluidstructure interaction is a rich and active field of mathematics that studies the interaction between fluids and solid objects. In this short article, we give a glimpse into this exciting field, as well as a ... 
Mathematics plays a key role in scientific computing
[SNAP2017011EN] (Mathematisches Forschungsinstitut Oberwolfach, 20171229)I attended a very interesting workshop at the research center MFO in Oberwolfach on “Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws”. The title sounds a bit technical, but in plain language ... 
Mathematische Modellierung von Krebswachstum
[SNAP2017004DE] (Mathematisches Forschungsinstitut Oberwolfach, 20171017)Krebs ist eine der größten Herausforderungen der modernen Medizin. Der WHO zufolge starben 2012 weltweit 8,2 Millionen Menschen an Krebs. Bis heute sind dessen molekulare Mechanismen nur in Teilen verstanden, was eine ... 
Matrixfaktorisierungen
[SNAP2014002DE] (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
[SNAP2015002EN] (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
[SNAP2018014EN] (Mathematisches Forschungsinstitut Oberwolfach, 20181229)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 ... 
Mixeddimensional models for realworld applications
[SNAP2019014EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191121)We explore mathematical models for physical problems in which it is necessary to simultaneously consider equations in different dimensions; these are called mixeddimensional models. We first give several examples, and ...