Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 23 Jun 2021 23:32:02 GMT2021-06-23T23:32:02ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
Diophantine Approximation in Metric Space
http://publications.mfo.de/handle/mfo/3864
Diophantine Approximation in Metric Space
Fraser, Jonathan M.; Koivusalo, Henna; Ramírez, Felipe A.
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of “well-spread” points, which we refer to as $abstract$ $rationals$. We prove various Jarník–Besicovitch type dimension bounds and investigate their sharpness
Mon, 14 Jun 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38642021-06-14T00:00:00ZFraser, Jonathan M.Koivusalo, HennaRamírez, Felipe A.Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of “well-spread” points, which we refer to as $abstract$ $rationals$. We prove various Jarník–Besicovitch type dimension bounds and investigate their sharpnessInvitation 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.On the Computational Content of the Theory of Borel Equivalence Relations
http://publications.mfo.de/handle/mfo/3849
On the Computational Content of the Theory of Borel Equivalence Relations
Bazhenov, Nikolay; Monin, Benoit; San Mauro, Luca; Zamora, Rafael
This preprint offers computational insights into the theory of Borel equivalence relations. Specifically, we classify equivalence relations on the Cantor space up to computable reductions, i.e., reductions induced by Turing functionals. The presented results correspond to three main research focuses: (i) the poset of degrees of equivalence relations on reals under computable reducibility; (ii) the complexity of the equivalence relations generated by computability-theoretic reducibilities $(\leqslant_T , \leqslant_{tt} , \leqslant_m , \leqslant_1 )$, (iii) the effectivization of the notion of hyperfiniteness.
Wed, 17 Mar 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38492021-03-17T00:00:00ZBazhenov, NikolayMonin, BenoitSan Mauro, LucaZamora, RafaelThis preprint offers computational insights into the theory of Borel equivalence relations. Specifically, we classify equivalence relations on the Cantor space up to computable reductions, i.e., reductions induced by Turing functionals. The presented results correspond to three main research focuses: (i) the poset of degrees of equivalence relations on reals under computable reducibility; (ii) the complexity of the equivalence relations generated by computability-theoretic reducibilities $(\leqslant_T , \leqslant_{tt} , \leqslant_m , \leqslant_1 )$, (iii) the effectivization of the notion of hyperfiniteness.The Elser Nuclei Sum Revisited
http://publications.mfo.de/handle/mfo/3846
The Elser Nuclei Sum Revisited
Grinberg, Darij
Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ \textit{pandemic} if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq\varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and a refinement using discrete Morse theory.
Tue, 16 Mar 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38462021-03-16T00:00:00ZGrinberg, DarijFix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ \textit{pandemic} if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq\varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and a refinement using discrete Morse theory.The C-Map as a Functor on Certain Variations of Hodge Structure
http://publications.mfo.de/handle/mfo/3845
The C-Map as a Functor on Certain Variations of Hodge Structure
Mantegazza, Mauro; Saha, Arpan
We give a new manifestly natural presentation of the supergravity c-map. We achieve this by giving a more explicit description of the correspondence between projective special Kähler manifolds and variations of Hodge structure, and by demonstrating that the twist construction of Swann, for a certain kind of twist data, reduces to a quotient by a discrete group. We combine these two ideas by showing that variations of Hodge structure give rise to the aforementioned kind of twist data and by then applying the twist realisation of the c-map due to Macia and Swann. This extends previous results regarding the lifting, along the c-map, of infinitesimal automorphisms to the lifting of general isomorphisms.
Mon, 15 Mar 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38452021-03-15T00:00:00ZMantegazza, MauroSaha, ArpanWe give a new manifestly natural presentation of the supergravity c-map. We achieve this by giving a more explicit description of the correspondence between projective special Kähler manifolds and variations of Hodge structure, and by demonstrating that the twist construction of Swann, for a certain kind of twist data, reduces to a quotient by a discrete group. We combine these two ideas by showing that variations of Hodge structure give rise to the aforementioned kind of twist data and by then applying the twist realisation of the c-map due to Macia and Swann. This extends previous results regarding the lifting, along the c-map, of infinitesimal automorphisms to the lifting of general isomorphisms.$C^*$-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.Amorphic Complexity of Group Actions with Applications to Quasicrystals
http://publications.mfo.de/handle/mfo/3830
Amorphic Complexity of Group Actions with Applications to Quasicrystals
Fuhrmann, Gabriel; Gröger, Maik; Jäger, Tobias; Kwietniak, Dominik
In this article, we define amorphic complexity for actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb Z$-actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via Meyer's cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.
Tue, 02 Feb 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38302021-02-02T00:00:00ZFuhrmann, GabrielGröger, MaikJäger, TobiasKwietniak, DominikIn this article, we define amorphic complexity for actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb Z$-actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We show that it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via Meyer's cut and project method. We provide sharp upper bounds on amorphic complexity of such systems. In doing so, we observe an intimate relationship between amorphic complexity and fractal geometry.