2021
2021
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Braid groups, the Yang–Baxter equation, and subfactors
[SNAP2021005EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021)The Yang–Baxter equation is a fascinating equation that appears in many areas of physics and mathematics and is best represented diagramatically. This snapshot connects the mathematics of braiding hair to the Yang–Baxter ... 
Describing distance: from the plane to spectral triples
[SNAP2021009EN] (Mathematisches Forschungsinstitut Oberwolfach, 20211231)Geometry 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 ... 
Finite geometries: pure mathematics close to applications
[SNAP2021010EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210922)The research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these ... 
Lagrangian mean curvature flow
[SNAP2021011EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210916)Lagrangian mean curvature flow is a powerful tool in modern mathematics with connections to topics in analysis, geometry, topology and mathematical physics. I will describe some of the key aspects of Lagrangian mean curvature ... 
Reflections on hyperbolic space
[SNAP2021007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210824)In school, we learn that the interior angles of any triangle sum up to pi. However, there exist spaces different from the usual Euclidean space in which this is not true. One of these spaces is the ''hyperbolic space'', ... 
The Enigma behind the Good–Turing formula
[SNAP2021008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210716)Finding the total number of species in a population based on a finite sample is a difficult but practically important problem. In this snapshot, we will attempt to shed light on how during World War II, two cryptanalysts, ... 
Ultrafilter methods in combinatorics
[SNAP2021006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210625)Given 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 ... 
Zopfgruppen, die Yang–BaxterGleichung und Unterfaktoren
[SNAP2021005DE] (Mathematisches Forschungsinstitut Oberwolfach, 20210624)Die Yang–BaxterGleichung 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 ... 
Invitation to quiver representation and Catalan combinatorics
[SNAP2021004EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210408)Representation theory is an area of mathematics that deals with abstract algebraic structures and has numerous applications across disciplines. In this snapshot, we will talk about the representation theory of a class ... 
Searching for structure in complex data: a modern statistical quest
[SNAP2021003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210329)Current research in statistics has taken interesting new directions, as data collected from scientific studies has become increasingly complex. At first glance, the number of experiments conducted by a scientist must ... 
$C^*$algebras: structure and classification
[SNAP2021002EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210223)The theory of $C^*$algebras traces its origins back to the development of quantum mechanics and it has evolved into a large and highly active field of mathematics. Much of the progress over the last couple of decades ... 
From the dollar game to the RiemannRoch Theorem
[SNAP2021001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210223)What is the dollar game? What can you do to win it? Can you always win it? In this snapshot you will find answers to these questions as well as several of the mathematical surprises that lurk in the background, including ...