2 - Snapshots of Modern Mathematics from Oberwolfach
http://publications.mfo.de/handle/mfo/20
The snapshot project is designed to promote the understanding and appreciation of modern mathematics and mathematical research in the general public world-wide. It is part of the project "Oberwolfach meets IMAGINARY“, supported by the Klaus Tschira Foundation.Fri, 28 Jan 2022 15:10:08 GMT2022-01-28T15:10:08ZDescribing distance: from the plane to spectral triples
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 GMThttp://publications.mfo.de/handle/mfo/39122021-12-31T00:00:00ZArici, FrancescaMesland, 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.Finite geometries: pure mathematics close to applications
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 GMThttp://publications.mfo.de/handle/mfo/38892021-09-22T00:00:00ZStorme, LeoThe 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”.Lagrangian mean curvature flow
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 GMThttp://publications.mfo.de/handle/mfo/38842021-09-16T00:00:00ZLotay, 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.Reflections on hyperbolic space
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 GMThttp://publications.mfo.de/handle/mfo/38762021-08-24T00:00:00ZHaensch, AnnaIn 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.The Enigma behind the Good–Turing formula
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 GMThttp://publications.mfo.de/handle/mfo/38752021-07-16T00:00:00ZBalabdaoui, FadouaKulagina, YuliaFinding 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.Ultrafilter methods in combinatorics
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 GMThttp://publications.mfo.de/handle/mfo/38702021-06-25T00:00:00ZGoldbring, 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”.Zopfgruppen, die Yang–Baxter-Gleichung und Unterfaktoren
http://publications.mfo.de/handle/mfo/3872
Zopfgruppen, die Yang–Baxter-Gleichung und Unterfaktoren; Braid groups, the Yang–Baxter equation, and subfactors
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.; 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".
Thu, 24 Jun 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38722021-06-24T00:00:00ZLechner, GandalfDie 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.
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".Invitation to quiver representation and Catalan combinatorics
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 GMThttp://publications.mfo.de/handle/mfo/38532021-04-08T00:00:00ZRognerud, BaptisteRepresentation 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.Searching for structure in complex data: a modern statistical quest
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 GMThttp://publications.mfo.de/handle/mfo/38512021-03-29T00:00:00ZLoh, Po-LingCurrent 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.$C^*$-algebras: structure and classification
http://publications.mfo.de/handle/mfo/3841
$C^*$-algebras: structure and classification
Kerr, David
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 has been driven by an ambitious program
of classification launched by George A. Elliott in the
1980s, and just recently this project has succeeded
in achieving one of its central goals in an unexpectedly
dramatic fashion. This Snapshot aims to recount
some of the fundamental ideas at play.
Tue, 23 Feb 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38412021-02-23T00:00:00ZKerr, DavidThe 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 has been driven by an ambitious program
of classification launched by George A. Elliott in the
1980s, and just recently this project has succeeded
in achieving one of its central goals in an unexpectedly
dramatic fashion. This Snapshot aims to recount
some of the fundamental ideas at play.