Snapshots of Modern Mathematics from Oberwolfach
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Browsing Snapshots of Modern Mathematics from Oberwolfach by Subject "510"
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- Item4 = 2 × 2, or the Power of Even Integers in Fourier Analysis(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Negro, Giuseppe; Oliveira e Silva, DiogoWe 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 analysis with surprising applications to number theory and geometric measure theory.
- ItemA tale of three curves(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Balakrishnan, Jennifer S.In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.
- ItemThe adaptive finite element method(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Gallistl, DietmarComputer 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 hand, a larger number of unknowns can improve the precision of the simulation. The adaptive finite element method (AFEM) is an algorithm for optimizing the choice of parameters so accurate simulation results can be obtained with as little computational effort as possible.
- ItemAlgebra, matrices, and computers(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Detinko, Alla; Flannery, Dane; Hulpke, AlexanderWhat 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 rely on the theory of matrix groups and new methods for handling matrix groups in a computer.
- ItemThe Algebraic Statistics of an Oberwolfach Workshop(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Seigal, AnnaAlgebraic 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 polynomial equalities and inequalities. We explore the connection between algebra and statistics for some small statistical models.
- ItemAlgebras and Quantum Games(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Paulsen, Vern I.Everyone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out!
- ItemAnalogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964)(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Petitgirard, LoïcThroughout 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 in the period 1948–1964, and we give examples of the specific (analogue) mathematical instruments they conceived and used. These examples also illustrate how their analogue culture and practices faced the advent of the digital computer, which appeared at that time as a new instrument, full of promises.
- ItemAperiodic Order and Spectral Properties(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Baake, Michael; Damanik, David; Grimm, UwePeriodic 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 going to explore highly ordered structures that are substantially nonperiodic, or aperiodic. As we construct such structures, we will discover surprising connections to various branches of mathematics, materials science, and physics. Let us catch a glimpse into the inherent beauty of aperiodic order!
- ItemArrangements of lines(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Harbourne, Brian; Szemberg, TomaszWe discuss certain open problems in the context of arrangements of lines in the plane.
- ItemBilliards and flat surfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Davis, DianaBilliards, 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.
- ItemBiological shape analysis with geometric statistics and learning(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Utpala, Saiteja; Miolane, NinaThe 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 the key to unlocking outstanding mysteries in biomedicine. This snapshot introduces the mathematical framework of geometric statistics and learning and its applications to biomedicine.
- ItemChaos and chaotic fluid mixing(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Solomon, TomVery 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 equation that can be easily simulated, and with mixing in simple fluid flows.
- ItemCharacterizations of intrinsic volumes on convex bodies and convex functions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Mussnig, FabianIf we want to express the size of a two-dimensional 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 classical mathematical results. We also take a look at applications and new generalizations to the setting of functions.
- ItemClosed geodesics on surfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Dozier, BenjaminWe consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line paths, known as geodesics, in these geometries. We discuss the issue of counting closed geodesics; this is particularly rich for hyperbolic (negatively curved) surfaces.
- ItemClosed geodesics on surfaces and Riemannian manifolds(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Radeschi, MarcoGeodesics are special paths in surfaces and so-called 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 about these special paths, and the efforts to find closed geodesics.
- ItemThe codimension(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Lerario, AntonioIn 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.
- ItemComputational Optimal Transport(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Solomon, JustinOptimal 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; modern applications must cope with thousands or millions of these at a time. Here, we introduce the computational optimal transport problem and summarize recent ideas for achieving new heights in efficiency and scalability.
- ItemComputing the long term evolution of the solar system with geometric numerical integrators(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Fiorelli Vilmart, Shaula; Vilmart, GillesSimulating 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 same initial data yields the correct behavior. We explain the main ideas of how the evolution of the solar system can be computed over long times by taking advantage of so-called geometric numerical methods. Short sample codes are provided for the Sun–Earth–Moon system.
- ItemComputing with symmetries(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Roney-Dougal, Colva M.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.
- ItemConfiguration spaces and braid groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Jiménez Rolland, Rita; Xicoténcatl, Miguel A.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 these groups determine features of the spaces.