1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Fri, 25 Jun 2021 09:12:36 GMT2021-06-25T09:12:36ZDiophantine 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 sharpnessOn 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.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.Lifting Spectral Triples to Noncommutative Principal Bundles
http://publications.mfo.de/handle/mfo/3827
Lifting Spectral Triples to Noncommutative Principal Bundles
Schwieger, Kay; Wagner, Stefan
Given a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a
spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.
Mon, 11 Jan 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38272021-01-11T00:00:00ZSchwieger, KayWagner, StefanGiven a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a
spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.Boundary Conditions for Scalar Curvature
http://publications.mfo.de/handle/mfo/3824
Boundary Conditions for Scalar Curvature
Bär, Christian; Hanke, Bernhard
Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite $K$-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites à la Miao and the deformation of boundary conditions à la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.
Mon, 04 Jan 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38242021-01-04T00:00:00ZBär, ChristianHanke, BernhardBased on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite $K$-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites à la Miao and the deformation of boundary conditions à la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.Dynamics of Gravitational Collapse in the Axisymmetric Einstein-Vlasov System
http://publications.mfo.de/handle/mfo/3820
Dynamics of Gravitational Collapse in the Axisymmetric Einstein-Vlasov System
Ames, Ellery; Andréasson, Håkan; Rinne, Oliver
We numerically investigate the dynamcis near black hole formation of solutions to the Einstein-Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the (2+1)+1 formulation of the Einstein field equations in axisymmetry. Solutions are launched from generic type initial data and exhibit type I critical behaviour. In particular we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscillations, and appear to disperse. We prove that complete dispersal of the solution implies that it has nonpositive binding energy.
Tue, 15 Dec 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38202020-12-15T00:00:00ZAmes, ElleryAndréasson, HåkanRinne, OliverWe numerically investigate the dynamcis near black hole formation of solutions to the Einstein-Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the (2+1)+1 formulation of the Einstein field equations in axisymmetry. Solutions are launched from generic type initial data and exhibit type I critical behaviour. In particular we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscillations, and appear to disperse. We prove that complete dispersal of the solution implies that it has nonpositive binding energy.Octonion Polynomials with Values in a Subalgebra
http://publications.mfo.de/handle/mfo/3802
Octonion Polynomials with Values in a Subalgebra
Chapman, Adam
In this paper, we prove that given an octonion algebra $A$ over a field $F$, a subring $E \subseteq F$ and an octonion $E$-algebra $R$ inside $A$, the set $S$ of polynomials $f(x) \in A[x]$ satisfying $f(R) \subseteq R$ is an octonion $(S\cap F[x])$-algebra, under the assumption that either $\frac{1}{2} \in R$ or $\operatorname{char}(F) \neq 0$, and $R$ contains the standard generators of $A$ and their inverses.
The project was inspired by a question raised by Werner on whether integer-valued octonion polynomials over the reals form a nonassociative ring. We also prove that the polynomials $\frac{1}{p}(x^{p^2}-x)(x^p-x)$ for prime $p$ are integer-valued in the ring of polynomials $A[x]$ over any real nonsplit Cayley-Dickson algebra $A$.
Thu, 22 Oct 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38022020-10-22T00:00:00ZChapman, AdamIn this paper, we prove that given an octonion algebra $A$ over a field $F$, a subring $E \subseteq F$ and an octonion $E$-algebra $R$ inside $A$, the set $S$ of polynomials $f(x) \in A[x]$ satisfying $f(R) \subseteq R$ is an octonion $(S\cap F[x])$-algebra, under the assumption that either $\frac{1}{2} \in R$ or $\operatorname{char}(F) \neq 0$, and $R$ contains the standard generators of $A$ and their inverses.
The project was inspired by a question raised by Werner on whether integer-valued octonion polynomials over the reals form a nonassociative ring. We also prove that the polynomials $\frac{1}{p}(x^{p^2}-x)(x^p-x)$ for prime $p$ are integer-valued in the ring of polynomials $A[x]$ over any real nonsplit Cayley-Dickson algebra $A$.Homology and $K$-Theory of Torsion-Free Ample Groupoids and Smale Spaces
http://publications.mfo.de/handle/mfo/3800
Homology and $K$-Theory of Torsion-Free Ample Groupoids and Smale Spaces
Proietti, Valerio; Yamashita, Makoto
Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the $K$-groups of the groupoid C*-algebra when the groupoid has torsion-free stabilizers and satisfies the strong Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture by Meyer and Nest. For the unstable equivalence relation of a Smale space with totally disconnected stable sets, this spectral sequence shows Putnam’s homology groups on the second sheet.
Fri, 09 Oct 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38002020-10-09T00:00:00ZProietti, ValerioYamashita, MakotoGiven an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the $K$-groups of the groupoid C*-algebra when the groupoid has torsion-free stabilizers and satisfies the strong Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture by Meyer and Nest. For the unstable equivalence relation of a Smale space with totally disconnected stable sets, this spectral sequence shows Putnam’s homology groups on the second sheet.