Browsing 2  Snapshots of modern mathematics from Oberwolfach by Title
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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'', ... 
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. 
The Robinson–Schensted algorithm
[SNAP2022002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220506)I am going to describe the Robinson–Schensted algorithm which transforms a permutation of the numbers from 1 to n into a pair of combinatorial objects called “standard Young tableaux”. I will then say a little bit about a ... 
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. 
Route planning for bacteria
[SNAP2022012EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221208)Bacteria have been fascinating biologists since their discovery in the late 17th century. By analysing their movements, mathematical models have been developed as a tool to understand their behaviour. However, adapting ... 
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 ... 
Searching for the monster in the trees
[SNAP2022003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220413)The Monster finite simple group is almost unimaginably large, with about 8 × 1053 elements in it. Trying to understand such an immense object requires both theory and computer programs. In this snapshot, we discuss finite ... 
Seeing through rock with help from optimal transport
[SNAP2022004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220506)Geophysicists and mathematicians work together to detect geological structures located deep within the earth by measuring and interpreting echoes from manmade earthquakes. This inverse problem naturally involves the ... 
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 inverse problems with Bayes' theorem
[SNAP2022006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220905)The goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as ... 
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 ... 
Spaces of Riemannian metrics
[SNAP2017010ENSNAP2017010ES] (Mathematisches Forschungsinstitut Oberwolfach, 20171228)Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the ...