Oberwolfach Publications
http://publications.mfo.de:80
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sun, 01 Aug 2021 19:16:41 GMT2021-08-01T19:16:41ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:80
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.Weak*-Continuity of Invariant Means on Spaces of Matrix Coefficients
http://publications.mfo.de/handle/mfo/3873
Weak*-Continuity of Invariant Means on Spaces of Matrix Coefficients
de Laat, Tim; Zadeh, Safoura
With every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ on $G$, each of which captures parts of the representation theory of $G$. Under certain natural assumptions, such a space $X(G)$ carries a unique invariant mean and has a natural predual, and we view the weak$^*$-continuity of this mean as a rigidity property of $G$. Important examples of such spaces $X(G)$, which we study explicitly, are the algebra $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Figà-Talamanca-Herz algebra $A_p(G)$ and the $p$-Fourier-Stieltjes algebra $B_p(G)$. In the setting of connected Lie groups $G$, we relate the weak$^*$-continuity of the mean on these spaces to structural properties of $G$. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.
Tue, 13 Jul 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38732021-07-13T00:00:00Zde Laat, TimZadeh, SafouraWith every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ on $G$, each of which captures parts of the representation theory of $G$. Under certain natural assumptions, such a space $X(G)$ carries a unique invariant mean and has a natural predual, and we view the weak$^*$-continuity of this mean as a rigidity property of $G$. Important examples of such spaces $X(G)$, which we study explicitly, are the algebra $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Figà-Talamanca-Herz algebra $A_p(G)$ and the $p$-Fourier-Stieltjes algebra $B_p(G)$. In the setting of connected Lie groups $G$, we relate the weak$^*$-continuity of the mean on these spaces to structural properties of $G$. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.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".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.