Snapshots of Modern Mathematics from Oberwolfach
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- 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.
- 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.
- 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.
- 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.
- ItemCounting self-avoiding walks on the hexagonal lattice(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Duminil-Copin, HugoIn how many ways can you go for a walk along a lattice grid in such a way that you never meet your own trail? In this snapshot, we describe some combinatorial and statistical aspects of these so-called self-avoiding walks. In particular, we discuss a recent result concerning the number of self-avoiding walks on the hexagonal (“honeycomb”) lattice. In the last part, we briefly hint at the connection to the geometry of long random self-avoiding walks.
- ItemCurriculum development in university mathematics: where mathematicians and education collide(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Sangwin, Christopher J.This snapshot looks at educational aspects of the design of curricula in mathematics. In particular, we examine choices textbook authors have made when introducing the concept of the completness of the real numbers. Can significant choices really be made? Do these choices have an effect on how people learn, and, if so, can we understand what they are?
- ItemC∗ -algebras: structure and classification(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Kerr, DavidThe theory of C∗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 has been driven by an ambitious program of classification launched by George A. Elliott in the 1980s, and just recently this project has succeeded in achieving one of its central goals in an unexpectedly dramatic fashion. This Snapshot aims to recount some of the fundamental ideas at play.
- ItemDarcy's law and groundwater flow modelling(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Schweizer, BenFormulations of natural phenomena are derived, sometimes, from experimentation and observation. Mathematical methods can be applied to expand on these formulations, and develop them into better models. In the year 1856, the French hydraulic engineer Henry Darcy performed experiments, measuring water flow through a column of sand. He discovered and described a fundamental law: the linear relation between pressure difference and flow rate – known today as Darcy’s law. We describe the law and the evolution of its modern formulation. We furthermore sketch some current mathematical research related to Darcy’s law.
- ItemData assimilation: mathematics for merging models and data(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Morzfeld, Matthias; Reich, SebastianWhen you describe a physical process, for example, the weather on Earth, or an engineered system, such as a self-driving car, you typically have two sources of information. The first is a mathematical model, and the second is information obtained by collecting data. To make the best predictions for the weather, or most effectively operate the self-driving car, you want to use both sources of information. Data assimilation describes the mathematical, numerical and computational framework for doing just that.
- ItemDeep Learning and Inverse Problems(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Arridge, Simon; de Hoop, Maarten; Maass, Peter; Öktem, Ozan; Schönlieb, Carola; Unser, MichaelBig data and deep learning are modern buzz words which presently infiltrate all fields of science and technology. These new concepts are impressive in terms of the stunning results they achieve for a large variety of applications. However, the theoretical justification for their success is still very limited. In this snapshot, we highlight some of the very recent mathematical results that are the beginnings of a solid theoretical foundation for the subject.