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.Sun, 01 Aug 2021 19:29:02 GMT2021-08-01T19:29:02ZThe 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.From the dollar game to the Riemann-Roch Theorem
http://publications.mfo.de/handle/mfo/3840
From the dollar game to the Riemann-Roch Theorem
Lamboglia, Sara; Ulirsch, Martin
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 a new perspective on a century-old theorem.
Tue, 23 Feb 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38402021-02-23T00:00:00ZLamboglia, SaraUlirsch, MartinWhat 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 a new perspective on a century-old theorem.Quantum symmetry
http://publications.mfo.de/handle/mfo/3831
Quantum symmetry
Caspers, Martijn
The 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.
Thu, 31 Dec 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38312020-12-31T00:00:00ZCaspers, 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.Shape space – a paradigm for character animation in computer graphics
http://publications.mfo.de/handle/mfo/3798
Shape space – a paradigm for character animation in computer graphics
Heeren, Behrend; Rumpf, Martin
Nowadays 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.
Wed, 07 Oct 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37982020-10-07T00:00:00ZHeeren, BehrendRumpf, 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.Higgs bundles without geometry
http://publications.mfo.de/handle/mfo/3793
Higgs bundles without geometry
Rayan, Steven; Schaposnik, Laura P.
Higgs bundles appeared a few decades ago as solutions
to certain equations from physics and have attracted
much attention in geometry as well as other
areas of mathematics and physics. Here, we take a
very informal stroll through some aspects of linear
algebra that anticipate the deeper structure in the
moduli space of Higgs bundles.
Tue, 29 Sep 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37932020-09-29T00:00:00ZRayan, StevenSchaposnik, Laura P.Higgs bundles appeared a few decades ago as solutions
to certain equations from physics and have attracted
much attention in geometry as well as other
areas of mathematics and physics. Here, we take a
very informal stroll through some aspects of linear
algebra that anticipate the deeper structure in the
moduli space of Higgs bundles.