http://publications.mfo.de/handle/mfo/3832 Tue, 07 Apr 2026 06:23:20 GMT 2026-04-07T06:23:20Z http://publications.mfo.de/handle/mfo/4126 Braid groups, the Yang–Baxter equation, and subfactors Lechner, Gandalf 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 equation and relates it to current research about systems of infinite dimensional algebras called "subfactors''. Fri, 01 Jan 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/4126 2021-01-01T00:00:00Z Lechner, Gandalf 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 equation and relates it to current research about systems of infinite dimensional algebras called "subfactors''. http://publications.mfo.de/handle/mfo/3912 Describing distance: from the plane to spectral triples Arici, Francesca; Mesland, Bram 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 glimpse into how certain “curved spaces” called manifolds can be better understood by looking at the (complex) differentiable functions they admit. Fri, 31 Dec 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3912 2021-12-31T00:00:00Z Arici, Francesca Mesland, Bram 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 glimpse into how certain “curved spaces” called manifolds can be better understood by looking at the (complex) differentiable functions they admit. http://publications.mfo.de/handle/mfo/3889 Finite geometries: pure mathematics close to applications Storme, Leo 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 structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”. Wed, 22 Sep 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3889 2021-09-22T00:00:00Z Storme, Leo 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 structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”. http://publications.mfo.de/handle/mfo/3884 Lagrangian mean curvature flow Lotay, Jason D. 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 flow, some recent progress, and some major open problems. Thu, 16 Sep 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3884 2021-09-16T00:00:00Z Lotay, Jason D. 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 flow, some recent progress, and some major open problems. http://publications.mfo.de/handle/mfo/3876 Reflections on hyperbolic space Haensch, Anna 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'', which has another geometry than the classical Euclidean geometry. In this snapshot, we consider the geometry of hyperbolic polytopes, for example polygons, how they tile hyperbolic space, and how reflections along the faces of polytopes give rise to important mathematical structures. The classification of these structures is an open area of research. Tue, 24 Aug 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3876 2021-08-24T00:00:00Z Haensch, Anna 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'', which has another geometry than the classical Euclidean geometry. In this snapshot, we consider the geometry of hyperbolic polytopes, for example polygons, how they tile hyperbolic space, and how reflections along the faces of polytopes give rise to important mathematical structures. The classification of these structures is an open area of research. http://publications.mfo.de/handle/mfo/3875 The Enigma behind the Good–Turing formula Balabdaoui, Fadoua; Kulagina, Yulia 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, Irving J. Good and Alan M. Turing, discovered one of the most widely applied formulas in statistics. The formula estimates the probability of missing some of the species in a sample drawn from a heterogeneous population. We will provide some intuition behind the formula, show its wide range of applications, and give a few technical details. Fri, 16 Jul 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3875 2021-07-16T00:00:00Z Balabdaoui, Fadoua Kulagina, Yulia 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, Irving J. Good and Alan M. Turing, discovered one of the most widely applied formulas in statistics. The formula estimates the probability of missing some of the species in a sample drawn from a heterogeneous population. We will provide some intuition behind the formula, show its wide range of applications, and give a few technical details. http://publications.mfo.de/handle/mfo/3870 Ultrafilter methods in combinatorics Goldbring, Isaac 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 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”. Fri, 25 Jun 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3870 2021-06-25T00:00:00Z Goldbring, Isaac 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 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”. http://publications.mfo.de/handle/mfo/3872 Zopfgruppen, die Yang–Baxter-Gleichung und Unterfaktoren Lechner, Gandalf 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.; [Also available in English] Thu, 24 Jun 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3872 2021-06-24T00:00:00Z Lechner, Gandalf 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. [Also available in English] http://publications.mfo.de/handle/mfo/3853 Invitation to quiver representation and Catalan combinatorics Rognerud, Baptiste 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 of objects called quivers and relate them to the fantastic combinatorics of the Catalan numbers. Thu, 08 Apr 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3853 2021-04-08T00:00:00Z Rognerud, Baptiste 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 of objects called quivers and relate them to the fantastic combinatorics of the Catalan numbers. http://publications.mfo.de/handle/mfo/3851 Searching for structure in complex data: a modern statistical quest Loh, Po-Ling 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 be fairly large in order for a statistician to draw correct conclusions based on noisy measurements of a large number of factors. However, statisticians may often uncover simpler structure in the data, enabling accurate statistical inference based on relatively few experiments. In this snapshot, we will introduce the concept of high-dimensional statistical estimation via optimization, and illustrate this principle using an example from medical imaging. We will also present several open questions which are actively being studied by researchers in statistics. Mon, 29 Mar 2021 00:00:00 GMT http://publications.mfo.de/handle/mfo/3851 2021-03-29T00:00:00Z Loh, Po-Ling 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 be fairly large in order for a statistician to draw correct conclusions based on noisy measurements of a large number of factors. However, statisticians may often uncover simpler structure in the data, enabling accurate statistical inference based on relatively few experiments. In this snapshot, we will introduce the concept of high-dimensional statistical estimation via optimization, and illustrate this principle using an example from medical imaging. We will also present several open questions which are actively being studied by researchers in statistics.