2 - Snapshots of Modern Mathematics from Oberwolfach
http://publications.mfo.de/handle/mfo/20
The snapshot project is designed to promote the understanding and appreciation of modern mathematics and mathematical research in the general public world-wide. It is part of the project "Oberwolfach meets IMAGINARY“, supported by the Klaus Tschira Foundation.Wed, 21 Feb 2024 18:35:33 GMT2024-02-21T18:35:33Z4 = 2 × 2, or the Power of Even Integers in Fourier Analysis
http://publications.mfo.de/handle/mfo/4105
4 = 2 × 2, or the Power of Even Integers in Fourier Analysis
Negro, Giuseppe; Oliveira e Silva, Diogo
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 analysis with surprising applications to number theory and geometric measure theory.
Sat, 30 Dec 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/41052023-12-30T00:00:00ZNegro, GiuseppeOliveira 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.Cutoff Phenomenon: Surprising Behaviour in Card Shuffling and other Markov Chains
http://publications.mfo.de/handle/mfo/4094
Cutoff Phenomenon: Surprising Behaviour in Card Shuffling and other Markov Chains
Baraquin, Isabelle; Lafrenière, Nadia; Schuh, Katharina
This snapshot compares two techniques of shuffling a deck of cards, asking how long it will take to shuffle the cards until a “well-mixed deck” is obtained. Surprisingly, the number of shuffles can be very different for very similar looking shuffling techniques.
Thu, 21 Dec 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40942023-12-21T00:00:00ZBaraquin, IsabelleLafrenière, NadiaSchuh, KatharinaThis snapshot compares two techniques of shuffling a deck of cards, asking how long it will take to shuffle the cards until a “well-mixed deck” is obtained. Surprisingly, the number of shuffles can be very different for very similar looking shuffling techniques.Algebras and Quantum Games
http://publications.mfo.de/handle/mfo/4089
Algebras and Quantum Games
Paulsen, Vern I.
Everyone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out!
Tue, 28 Nov 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40892023-11-28T00:00:00ZPaulsen, Vern I.Everyone loves a good game, but when the players can access the counterintuitive world of quantum mechanics, watch out!Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften
http://publications.mfo.de/handle/mfo/4069
Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften
Saberi, Ingmar
Wir werden einige grundlegende Ideen der Eichtheorie und der dazugehörigen Differentialtopologie erkunden. Damit kann sich die Leserin ein Bild des Modulraums flacher Zusammenhänge machen und ihn mit den physikalisch motivierten Ideen dahinter in Beziehung bringen. Den Begriffen von Symmetrien und Feldern gehen wir gründlich nach. Außerdem werfen wir einen flüchtigen Blick auf unendliche Symmetrie in zwei Dimensionen und auf vor kurzem entdeckte Verallgemeinerungen.
Tue, 19 Sep 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40692023-09-19T00:00:00ZSaberi, IngmarWir werden einige grundlegende Ideen der Eichtheorie und der dazugehörigen Differentialtopologie erkunden. Damit kann sich die Leserin ein Bild des Modulraums flacher Zusammenhänge machen und ihn mit den physikalisch motivierten Ideen dahinter in Beziehung bringen. Den Begriffen von Symmetrien und Feldern gehen wir gründlich nach. Außerdem werfen wir einen flüchtigen Blick auf unendliche Symmetrie in zwei Dimensionen und auf vor kurzem entdeckte Verallgemeinerungen.The Periodic Tables of Algebraic Geometry
http://publications.mfo.de/handle/mfo/4067
The Periodic Tables of Algebraic Geometry
Belmans, Pieter
To understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use in physics, chemistry, and biology. In mathematics, and algebraic geometry in particular, there are many instances of similar “periodic tables”, describing fundamental classification results. We will go on a tour of some of these.
Mon, 04 Sep 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40672023-09-04T00:00:00ZBelmans, PieterTo understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use in physics, chemistry, and biology. In mathematics, and algebraic geometry in particular, there are many instances of similar “periodic tables”, describing fundamental classification results. We will go on a tour of some of these.Patterns and Waves in Theory, Experiment, and Application
http://publications.mfo.de/handle/mfo/4053
Patterns and Waves in Theory, Experiment, and Application
Bramburger, Jason J.
In this snapshot of modern mathematics we describe some of the most prevalent waves and patterns that can arise in mathematical models and which are used to describe a number of biological, chemical, physical, and social processes. We begin by focussing on two types of patterns that do not change in time: space-filling patterns and localized patterns. We then discuss two types of waves that evolve predictably as time goes on: spreading waves and rotating waves. All our examples are motivated with real-world applications and we highlight some of the main lines of research that mathematicians pursue to better understand them.
Tue, 04 Jul 2023 00:00:00 GMThttp://publications.mfo.de/handle/mfo/40532023-07-04T00:00:00ZBramburger, Jason J.In this snapshot of modern mathematics we describe some of the most prevalent waves and patterns that can arise in mathematical models and which are used to describe a number of biological, chemical, physical, and social processes. We begin by focussing on two types of patterns that do not change in time: space-filling patterns and localized patterns. We then discuss two types of waves that evolve predictably as time goes on: spreading waves and rotating waves. All our examples are motivated with real-world applications and we highlight some of the main lines of research that mathematicians pursue to better understand them.Closed geodesics on surfaces
http://publications.mfo.de/handle/mfo/3998
Closed geodesics on surfaces
Dozier, Benjamin
We 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.
Thu, 08 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39982022-12-08T00:00:00ZDozier, 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.Route planning for bacteria
http://publications.mfo.de/handle/mfo/3997
Route planning for bacteria
Hellmuth, Kathrin; Klingenberg, Christian
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 these models to real situations can be challenging, because the model coefficients cannot be observed directly. In this snapshot, we study this question mathematically and explain how the idea of “route planning” can be used to determine these model coefficients.
Thu, 08 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39972022-12-08T00:00:00ZHellmuth, KathrinKlingenberg, ChristianBacteria 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 these models to real situations can be challenging, because the model coefficients cannot be observed directly. In this snapshot, we study this question mathematically and explain how the idea of “route planning” can be used to determine these model coefficients.Characterizations of intrinsic volumes on convex bodies and convex functions
http://publications.mfo.de/handle/mfo/3996
Characterizations of intrinsic volumes on convex bodies and convex functions; Charakterisierungen von inneren Volumina auf konvexen Körpern und konvexen Funktionen
Mussnig, Fabian
If 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.; Wenn wir die Größe einer zweidimensionalen Form mittels einer Zahl ausdrücken wollen, dann denken wir gewöhnlich an ihren Flächeninhalt oder ihren Umfang. Aber was macht diese Kennzahlen so besonders? Wir beantworten diese Frage anhand klassischer mathematischer Resultate und werfen einen Blick auf Anwendungen und Verallgemeinerungen dieser Theorie.
Thu, 08 Dec 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39962022-12-08T00:00:00ZMussnig, 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.
Wenn wir die Größe einer zweidimensionalen Form mittels einer Zahl ausdrücken wollen, dann denken wir gewöhnlich an ihren Flächeninhalt oder ihren Umfang. Aber was macht diese Kennzahlen so besonders? Wir beantworten diese Frage anhand klassischer mathematischer Resultate und werfen einen Blick auf Anwendungen und Verallgemeinerungen dieser Theorie.A tale of three curves
http://publications.mfo.de/handle/mfo/3986
A tale of three curves
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.
Thu, 27 Oct 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39862022-10-27T00:00:00ZBalakrishnan, 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.