Browsing 2  Snapshots of Modern Mathematics from Oberwolfach by Title
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Operator theory and the singular value decomposition
[SNAP2014009EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)This is a snapshot about operator theory and one of its fundamental tools: the singular value decomposition (SVD). The SVD breaks up linear transformations into simpler mappings, thus unveiling their geometric properties. ... 
Polyhedra and commensurability
[SNAP2016009EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)This snapshot introduces the notion of commensurability of polyhedra. At its bottom, this concept can be developed from constructions with paper, scissors, and glue. Starting with an elementary example, we formalize it ... 
Positive Scalar Curvature and Applications
[SNAP2019004ENSNAP2019004ES] (Mathematisches Forschungsinstitut Oberwolfach, 20190425)[also available in Spanish] 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 ... 
Prime tuples in function fields
[SNAP2016010EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)How many prime numbers are there? How are they distributed among other numbers? These are questions that have intrigued mathematicians since ancient times. However, many questions in this area have remained unsolved, and ... 
Das Problem der Kugelpackung
[SNAP2016004DE] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen ... 
Profinite groups
[SNAP2016014EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Profinite objects are mathematical constructions used to collect, in a uniform manner, facts about infinitely many finite objects. We shall review recent progress in the theory of profinite groups, due to Nikolov and Segal, ... 
Prony’s method: an old trick for new problems
[SNAP2018004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180306)In 1795, French mathematician Gaspard de Prony invented an ingenious trick to solve a recovery problem, aiming at reconstructing functions from their values at given points, which arose from a specific application in ... 
Quantum diffusion
[SNAP2015014EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)If you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see ... 
Quantum symmetry
[SNAP2020005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200604)In mathematics, symmetry is usually captured using the formalism of groups. However, the developments of the past few decades revealed the need to go beyond groups: to “quantum groups”. We explain the passage from ... 
Quantum symmetry
[SNAP2020009EN] (Mathematisches Forschungsinstitut Oberwolfach, 20201231)The symmetry of objects plays a crucial role in many branches of mathematics and physics. It allowed, for example, the early prediction of the existence of new small particles. “Quantum symmetry” concerns a generalized ... 
Random matrix theory: Dyson Brownian motion
[SNAP2020002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200415)The theory of random matrices was introduced by John Wishart (1898–1956) in 1928. The theory was then developed within the field of nuclear physics from 1955 by Eugene Paul Wigner (1902–1995) and later by Freeman John ... 
Random permutations
[SNAP2019007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190712)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
[SNAP2016002EN] (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 ... 
Reflections on hyperbolic space
[SNAP2021007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210824)In school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', ... 
Rotating needles, vibrating strings, and Fourier summation
[SNAP2020006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200921)We give a brief survey of the connection between seemingly unrelated problems such as sets in the plane containing lines pointing in many directions, vibrating strings and drum heads, and a classical problem from Fourier analysis. 
Searching for structure in complex data: a modern statistical quest
[SNAP2021003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210329)Current research in statistics has taken interesting new directions, as data collected from scientific studies has become increasingly complex. At first glance, the number of experiments conducted by a scientist must ... 
Shape space – a paradigm for character animation in computer graphics
[SNAP2020007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20201007)Nowadays 3D computer animation is increasingly realistic as the models used for the characters become more and more complex. These models are typically represented by meshes of hundreds of thousands or even millions ... 
A short story on optimal transport and its many applications
[SNAP2018013EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180808)We present some examples of optimal transport problems and of applications to different sciences (logistics, economics, image processing, and a little bit of evolution equations) through the crazy story of an industrial ... 
Snake graphs, perfect matchings and continued fractions
[SNAP2019001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190213)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 ... 
Solving quadratic equations in many variables
[SNAP2017012EN] (Mathematisches Forschungsinstitut Oberwolfach, 20171230)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 ...