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.Wed, 12 May 2021 04:21:22 GMT2021-05-12T04:21:22ZInvitation 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.Rotating needles, vibrating strings, and Fourier summation
http://publications.mfo.de/handle/mfo/3777
Rotating needles, vibrating strings, and Fourier summation
Zahl, Joshua
We give a brief survey of the connection between seemingly unrelated problems such as sets in the plane containing lines pointing in many directions, vibrating strings and drum heads, and a classical problem from Fourier analysis.
Mon, 21 Sep 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37772020-09-21T00:00:00ZZahl, JoshuaWe give a brief survey of the connection between seemingly unrelated problems such as sets in the plane containing lines pointing in many directions, vibrating strings and drum heads, and a classical problem from Fourier analysis.Quantum symmetry
http://publications.mfo.de/handle/mfo/3747
Quantum symmetry
Weber, Moritz
In mathematics, symmetry is usually captured using
the formalism of groups. However, the developments
of the past few decades revealed the need to go beyond
groups: to “quantum groups”. We explain the
passage from spaces to quantum spaces, from groups
to quantum groups, and from symmetry to quantum
symmetry, following an analytical approach.
Thu, 04 Jun 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37472020-06-04T00:00:00ZWeber, MoritzIn mathematics, symmetry is usually captured using
the formalism of groups. However, the developments
of the past few decades revealed the need to go beyond
groups: to “quantum groups”. We explain the
passage from spaces to quantum spaces, from groups
to quantum groups, and from symmetry to quantum
symmetry, following an analytical approach.Determinacy versus indeterminacy
http://publications.mfo.de/handle/mfo/3739
Determinacy versus indeterminacy
Berg, Christian
Can 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.
Wed, 22 Apr 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37392020-04-22T00:00:00ZBerg, 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.