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Closed geodesics on surfaces
[SNAP2022013EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221208)We consider surfaces of three types: the sphere, the torus, and manyholed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line ... 
Route planning for bacteria
[SNAP2022012EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221208)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 ... 
Characterizations of intrinsic volumes on convex bodies and convex functions
[SNAP2022011ENSNAP2022011DE] (Mathematisches Forschungsinstitut Oberwolfach, 20221208)If we want to express the size of a twodimensional 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 ... 
A tale of three curves
[SNAP2022010EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221027)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 ... 
What is pattern?
[SNAP2022009EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221025)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
[SNAP2022008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221025)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 ... 
Representations and degenerations
[SNAP2022007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20221025)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. 
Solving inverse problems with Bayes' theorem
[SNAP2022006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220905)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 ... 
Jewellery from tessellations of hyperbolic space
[SNAP2022005EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220602)In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a numbertheoretic origin give rise to a large variety of interesting tessellations of 3dimensional ... 
Seeing through rock with help from optimal transport
[SNAP2022004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220506)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 ... 
The Robinson–Schensted algorithm
[SNAP2022002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220506)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 ... 
Searching for the monster in the trees
[SNAP2022003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220413)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 ... 
Emergence in biology and social sciences
[SNAP2022001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220331)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 ...