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Submitted by IMAGINARY on
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.
Submitted by IMAGINARY on
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.
Submitted by IMAGINARY on
The Monster finite simple group is almost unimaginably large, with about 8×1053 elements in it. Trying to understand such an immense object requires both theory and computer programs. In this snapshot, we discuss finite groups, representations, and finally Brauer trees, which offer some new understanding of this vast and intricate structure.
Submitted by IMAGINARY on
I am going to describe the Robinson–Schensted algorithm which transforms a permutation of the numbers from 1 to n into a pair of combinatorial objects called “standard Young tableaux”. I will then say a little bit about a few of the fascinating properties of this transformation, and how it connects to current research.
Submitted by IMAGINARY on
Mathematics 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.
Submitted by IMAGINARY on
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.
Also available in German.
Submitted by IMAGINARY on
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.
Submitted by IMAGINARY on
We consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative cur- vature, 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.
Submitted by Bianca Violet on
“Future Mobility - The Board Game” is for 3-4 players aged 8 and older, made to be printed by yourself. Players move through their city applying concepts of real urban planning, such as the Avoid - Shift - Improve model.
Submitted by IMAGINARY on
Die Yang–Baxter-Gleichung ist eine faszinierende Gleichung, die in vielen Gebieten der Physik und der Mathematik auftritt und die am besten diagrammatisch dargestellt wird. Dieser Snapshot schlägt einen weiten Bogen vom Zöpfeflechten über die Yang–Baxter-Gleichung bis hin zur aktuellen Forschung zu Systemen von unendlichdimensionalen Algebren, die wir „Unterfaktoren“ nennen.