Browsing 2  Snapshots of Modern Mathematics from Oberwolfach by Title
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On the containment problem
[SNAP2016003EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Mathematicians routinely speak two languages: the language of geometry and the language of algebra. When translating between these languages, curves and lines become sets of polynomials called “ideals”. Often there are ... 
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. ... 
Patterns and Waves in Theory, Experiment, and Application
[SNAP2023001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20230704)In this snapshot of modern mathematics we describe some of the most prevalent waves and patterns that can arise in mathematical models and which are used to describe a number of biological, chemical, physical, and social ... 
The Periodic Tables of Algebraic Geometry
[SNAP2023002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20230904)To understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use ... 
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
[SNAP2019004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190425)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 why ... 
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 ... 
Le problème ternaire de Goldbach
[SNAP2014003FR] (Mathematisches Forschungsinstitut Oberwolfach, 2024)Leonhard Euler (1707–1783), l’un des plus grands mathématiciens du XVIIIe siècle et de tous les temps, entretenait une correspondance régulière avec l’un de ses amis: Christian Goldbach (1690–1764), un amateur polymathe ... 
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 ... 
Randomness is Natural  an Introduction to Regularisation by Noise
[SNAP2024002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20240522)Differential equations make predictions on the future state of a system given the present. In order to get a sensible prediction, sometimes it is necessary to include randomness in differential equations, taking microscopic ... 
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'', ... 
Representations and degenerations
[SNAP2022007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221025)In this snapshot, we explain two important mathematical concepts (representation and degeneration) in elementary terms. We will focus on the simplest meaningful examples, and motivate both concepts by study of symmetry.