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- ItemJewellery from tessellations of hyperbolic space(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Gangl, 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.
- ItemDescribing distance: from the plane to spectral triples(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Arici, Francesca; Mesland, BramGeometry draws its power from the abstract structures that govern the shapes found in the real world. These abstractions often provide deeper insights into the underlying mathematical objects. In this snapshot, we give a glimpse into how certain “curved spaces” called manifolds can be better understood by looking at the (complex) differentiable functions they admit.
- ItemSeeing through rock with help from optimal transport(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Frederick, Christina; Yang, 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.
- ItemUltrafilter methods in combinatorics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Goldbring, IsaacGiven a set X, ultrafilters determine which subsets of X should be considered as large. We illustrate the use of ultrafilter methods in combinatorics by discussing two cornerstone results in Ramsey theory, namely Ramsey’s theorem itself and Hindman’s theorem. We then present a recent result in combinatorial number theory that verifies a conjecture of Erdos known as the “B + C conjecture”.
- ItemShape space – a paradigm for character animation in computer graphics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Heeren, Behrend; Rumpf, MartinNowadays 3D computer animation is increasingly realistic as the models used for the characters become more and more complex. These models are typically represented by meshes of hundreds of thousands or even millions of triangles. The mathematical notion of a shape space allows us to effectively model, manipulate, and animate such meshes. Once an appropriate notion of dissimilarity measure between different triangular meshes is defined, various useful tools in character modeling and animation turn out to coincide with basic geometric operations derived from this definition.
- ItemSearching for the Monster in the Trees(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Craven, David A.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.
- ItemQuantum symmetry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Caspers, MartijnThe symmetry of objects plays a crucial role in many branches of mathematics and physics. It allowed, for example, the early prediction of the existence of new small particles. “Quantum symmetry” concerns a generalized notion of symmetry. It is an abstract way of characterizing the symmetry of a much richer class of mathematical and physical objects. In this snapshot we explain how quantum symmetry emerges as matrix symmetries using a famous example: Mermin’s magic square. It shows that quantum symmetries can solve problems that lie beyond the reach of classical symmetries, showing that quantum symmetries play a central role in modern mathematics.
- 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.
- ItemVertex-to-Self Trajectories on the Platonic Solids(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2020) Athreya, Jayadev S.; Aulicino, DavidWe consider the problem of walking in a straight line on the surface of a Platonic solid. While the tetrahedron, octahedron, cube, and icosahedron all exhibit the same behavior, we find a remarkable difference with the dodecahedron.
- ItemThe Robinson–Schensted algorithm(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Thomas, HughI 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.
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