Snapshots of Modern Mathematics from Oberwolfach
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Browsing Snapshots of Modern Mathematics from Oberwolfach by Subject "Analysis"
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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.
- 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!
- 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.
- 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.
- 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.
- ItemDeterminacy versus indeterminacy(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Berg, ChristianCan a continuous function on an interval be uniquely determined if we know all the integrals of the function against the natural powers of the variable? Following Weierstrass and Stieltjes, we show that the answer is yes if the interval is finite, and no if the interval is infinite.
- ItemDirichlet Series(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) McCarthy, John E.Mathematicians are very interested in prime numbers. In this snapshot, we will discuss some problems concerning the distribution of primes and introduce some special infinite series in order to study them.
- ItemEmergence in biology and social sciences(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Hoffmann, Franca; Merino-Aceituno, SaraMathematics is the key to linking scientific knowledge at different scales: from microscopic to macroscopic dynamics. This link gives us understanding on the emergence of observable patterns like flocking of birds, leaf venation, opinion dynamics, and network formation, to name a few. In this article, we explore how mathematics is able to traverse scales, and in particular its application in modelling collective motion of bacteria driven by chemical signalling.
- ItemExpander graphs and where to find them(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Khukhro, AnaGraphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours.
- ItemGeometry behind one of the Painlevé III differential equations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Hertling, ClausThe Painlevé equations are second order differential equations, which were first studied more than 100 years ago. Nowadays they arise in many areas in mathematics and mathematical physics. This snapshot discusses the solutions of one of the Painlevé equations and presents old results on the asymptotics at two singular points and new results on the global behavior.
- ItemThe Kadison-Singer problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Valette, AlainIn quantum mechanics, unlike in classical mechanics, one cannot make precise predictions about how a system will behave. Instead, one is concerned with mere probabilities. Consequently, it is a very important task to determine the basic probabilities associated with a given system. In this snapshot we will present a recent uniqueness result concerning these probabilities.
- ItemLimits of graph sequences(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Klimošová, TerezaGraphs are simple mathematical structures used to model a wide variety of real-life objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to efficiently represent and study properties of extremely large graphs led to the development of the theory of graph limits.
- ItemThe mathematics of aquatic locomotion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Tucsnak, MariusAquatic locomotion is a self-propelled motion through a liquid medium. It can be of biological nature (fish, microorganisms,. . .) or performed by robotic swimmers. This snapshot aims to introduce the reader to some of the challenges raised by the mathematical modelling of aquatic locomotion, even in seemingly very simple cases.
- ItemThe Mathematics of Fluids and Solids(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Kaltenbacher, Barbara; Kukavica, Igor; Lasiecka, Irena; Triggiani, Roberto; Tuffaha, Amjad; Webster, JustinFluid-structure interaction is a rich and active field of mathematics that studies the interaction between fluids and solid objects. In this short article, we give a glimpse into this exciting field, as well as a sample of the most significant questions that mathematicians try to answer.
- ItemMathematische Modellierung von Krebswachstum(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Engwer, Christian; Knappitsch, MarkusKrebs ist eine der größten Herausforderungen der modernen Medizin. Der WHO zufolge starben 2012 weltweit 8,2 Millionen Menschen an Krebs. Bis heute sind dessen molekulare Mechanismen nur in Teilen verstanden, was eine erfolgreiche Behandlung erschwert. Mathematische Modellierung und Computersimulationen können helfen, die Mechanismen des Tumorwachstums besser zu verstehen. Sie eröffnen somit neue Chancen für zukünftige Behandlungsmethoden. In diesem Schnappschuss steht die mathematische Modellierung von Glioblastomen im Fokus, einer Klasse sehr agressiver Tumore im menschlichen Gehirn.
- ItemMinimizing energy(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Breiner, ChristineWhat is the most efficient way to fence land when you’ve only got so many metres of fence? Or, to put it differently, what is the largest area bounded by a simple closed planar curve of fixed length? We consider the answer to this question and others like it, making note of recent results in the same spirit.