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.Mon, 27 Mar 2023 07:00:20 GMT2023-03-27T07:00:20ZClosed 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.What is pattern?
http://publications.mfo.de/handle/mfo/3983
What is pattern?
Baake, Michael; Grimm, Uwe; Moody, Robert V.
Pattern is ubiquitous and seems totally familiar. Yet if we ask what it is, we find a bewildering collection of answers. Here we suggest that there is a common thread, and it revolves around dynamics.
Tue, 25 Oct 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39832022-10-25T00:00:00ZBaake, MichaelGrimm, UweMoody, Robert V.Pattern is ubiquitous and seems totally familiar. Yet if we ask what it is, we find a bewildering collection of answers. Here we suggest that there is a common thread, and it revolves around dynamics.Biological shape analysis with geometric statistics and learning
http://publications.mfo.de/handle/mfo/3985
Biological shape analysis with geometric statistics and learning
Utpala, Saiteja; Miolane, Nina
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 the key to unlocking outstanding mysteries in biomedicine. This snapshot introduces the mathematical framework of geometric statistics and learning and its applications to biomedicine.
Tue, 25 Oct 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39852022-10-25T00:00:00ZUtpala, SaitejaMiolane, 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.Representations and degenerations
http://publications.mfo.de/handle/mfo/3984
Representations and degenerations
Dumanski, Ilya; Kiritchenko, Valentina
In this snapshot, we explain two important mathematical concepts (representation and degeneration) in elementary terms. We will focus on the simplest meaningful examples, and motivate both concepts by study of symmetry.
Tue, 25 Oct 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39842022-10-25T00:00:00ZDumanski, IlyaKiritchenko, ValentinaIn this snapshot, we explain two important mathematical concepts (representation and degeneration) in elementary terms. We will focus on the simplest meaningful examples, and motivate both concepts by study of symmetry.Solving inverse problems with Bayes' theorem
http://publications.mfo.de/handle/mfo/3972
Solving inverse problems with Bayes' theorem
Latz, Jonas; Sprungk, Björn
The goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as the Bayesian approach. In this approach, the unknown parameter is modelled as a random variable to reflect its uncertain value. Bayes’ theorem is applied to update our knowledge given new information from noisy data.
Mon, 05 Sep 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39722022-09-05T00:00:00ZLatz, JonasSprungk, BjörnThe goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as the Bayesian approach. In this approach, the unknown parameter is modelled as a random variable to reflect its uncertain value. Bayes’ theorem is applied to update our knowledge given new information from noisy data.Jewellery from tessellations of hyperbolic space
http://publications.mfo.de/handle/mfo/3952
Jewellery from tessellations of hyperbolic space
Gangl, Herbert
In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.
Thu, 02 Jun 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39522022-06-02T00:00:00ZGangl, HerbertIn this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.Seeing through rock with help from optimal transport
http://publications.mfo.de/handle/mfo/3941
Seeing through rock with help from optimal transport
Frederick, Christina; Yang, Yunan
Geophysicists and mathematicians work together to detect geological structures located deep within the earth by measuring and interpreting echoes from manmade earthquakes. This inverse problem naturally involves the mathematics of wave propagation, but we will see that a different mathematical theory – optimal transport – also turns out to be very useful.
Fri, 06 May 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39412022-05-06T00:00:00ZFrederick, ChristinaYang, YunanGeophysicists and mathematicians work together to detect geological structures located deep within the earth by measuring and interpreting echoes from manmade earthquakes. This inverse problem naturally involves the mathematics of wave propagation, but we will see that a different mathematical theory – optimal transport – also turns out to be very useful.