Browsing 2  Snapshots of Modern Mathematics from Oberwolfach by Title
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Footballs and donuts in four dimensions
[SNAP2016012EN] (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 ... 
Formation Control and Rigidity Theory
[SNAP2019017EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191211)Formation control is one of the fundamental coordination tasks for teams of autonomous vehicles. Autonomous formations are used in applications ranging from searchandrescue operations to deep space exploration, with ... 
Friezes and tilings
[SNAP2015004EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)Friezes have occured as architectural ornaments for many centuries. In this snapshot, we consider the mathematical analogue of friezes as introduced in the 1970s by Conway and Coxeter. Recently, infinite versions of such ... 
From Betti numbers to ℓ²Betti numbers
[SNAP2020001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200415)We provide a leisurely introduction to ℓ²Betti 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 ℓ²Betti numbers. 
From computer algorithms to quantum field theory: an introduction to operads
[SNAP2015017EN] (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 RiemannRoch Theorem
[SNAP2021001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210223)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
[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 ... 
The Geometry of Fair Division
[SNAP2023007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20231230)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 nongeometric problems. The strategy we develop to solve ... 
Geproci Sets: a New Perspective in Algebraic Geometry
[SNAP2023008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20231230)Geproci sets arise from applying the perspective of inverse scattering problems to algebraic geometry. Analogous to the reconstruction of an object from multiple Xray images, we aim at a classification of sets with certain ... 
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 stroll ... 
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
[SNAP2015009EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)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. The group might ask: what ... 
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 ...