Browsing 2  Snapshots of Modern Mathematics from Oberwolfach by Title
Now showing items 120 of 131

4 = 2 × 2, or the Power of Even Integers in Fourier Analysis
[SNAP2023006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20231230)We describe how simple observations related to vectors of length 1 recently led to the proof of an important mathematical fact: the sharp Stein–Tomas inequality from Fourier restriction theory, a pillar of modern harmonic ... 
The adaptive finite element method
[SNAP2016013EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Computer simulations of many physical phenomena rely on approximations by models with a finite number of unknowns. The number of these parameters determines the computational effort needed for the simulation. On the other ... 
Algebra, matrices, and computers
[SNAP2019005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190503)What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that ... 
The Algebraic Statistics of an Oberwolfach Workshop
[SNAP2018001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180227)Algebraic Statistics builds on the idea that statistical models can be understood via polynomials. Many statistical models are parameterized by polynomials in the model parameters; others are described implicitly by ... 
Algebras and Quantum Games
[SNAP2023004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20231128)Everyone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out! 
Analogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964)
[SNAP2019012EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191121)Throughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers ... 
Aperiodic Order and Spectral Properties
[SNAP2017003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20170914)Periodic structures like a typical tiled kitchen floor or the arrangement of carbon atoms in a diamond crystal certainly possess a high degree of order. But what is order without periodicity? In this snapshot, we are ... 
Arrangements of lines
[SNAP2014005EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)We discuss certain open problems in the context of arrangements of lines in the plane. 
Billiards and flat surfaces
[SNAP2015001ENSNAP2015001DE] (Mathematisches Forschungsinstitut Oberwolfach, 2015)[also available in German] Billiards, the study of a ball bouncing around on a table, is a rich area of current mathematical research. We discuss questions and results on billiards, and on the related topic of flat surfaces. 
Biological shape analysis with geometric statistics and learning
[SNAP2022008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221025)The advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold ... 
$C^*$algebras: structure and classification
[SNAP2021002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210223)The theory of $C^*$algebras traces its origins back to the development of quantum mechanics and it has evolved into a large and highly active field of mathematics. Much of the progress over the last couple of decades ... 
Chaos and chaotic fluid mixing
[SNAP2015005EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)Very simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with behavior that is sensitive to small deviations in the initial conditions. We illustrate this with a single recurrence ... 
Characterizations of intrinsic volumes on convex bodies and convex functions
[SNAP2022011ENSNAP2022011DE] (Mathematisches Forschungsinstitut Oberwolfach, 20221208)If we want to express the size of a twodimensional shape with a number, then we usually think about its area or circumference. But what makes these quantities so special? We give an answer to this question in terms of ... 
Closed geodesics on surfaces
[SNAP2022013EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221208)We consider surfaces of three types: the sphere, the torus, and manyholed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line ... 
Closed geodesics on surfaces and Riemannian manifolds
[SNAP2017005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20171207)Geodesics are special paths in surfaces and socalled Riemannian manifolds which connect close points in the shortest way. Closed geodesics are geodesics which go back to where they started. In this snapshot we talk ... 
The codimension
[SNAP2018009EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180619)In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one. 
Computational Optimal Transport
[SNAP2017008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20171221)Optimal transport is the mathematical discipline of matching supply to demand while minimizing shipping costs. This matching problem becomes extremely challenging as the quantity of supply and demand points increases; ... 
Computing the long term evolution of the solar system with geometric numerical integrators
[SNAP2017009EN] (Mathematisches Forschungsinstitut Oberwolfach, 20171227)Simulating the dynamics of the Sun–Earth–Moon system with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the ... 
Computing with symmetries
[SNAP2018003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180306)Group theory is the study of symmetry, and has many applications both within and outside mathematics. In this snapshot, we give a brief introduction to symmetries, and how to compute with them. 
Configuration spaces and braid groups
[SNAP2019011EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191008)In this snapshot we introduce configuration spaces and explain how a mathematician studies their ‘shape’. This will lead us to consider paths of configurations and braid groups, and to explore how algebraic properties of ...