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, 05 Mar 2021 21:31:26 GMT2021-03-05T21:31:26ZAmorphic 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.Unexpected Properties of the Klein Configuration of 60 Points in $\mathbb{P}^3$
http://publications.mfo.de/handle/mfo/3799
Unexpected Properties of the Klein Configuration of 60 Points in $\mathbb{P}^3$
Pokora, Piotr; Szemberg, Tomasz; Szpond, Justyna
Felix Klein in course of his study of the regular and its symmetries encountered a highly symmetric configuration of 60 points in $\mathbb{P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the 60 reflection planes in the group $G_{31}$ in the Shephard-Todd list.
In the present note we show that the 60 points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree 6. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of 60 points is a cone with a single singularity of multiplicity 6 and the other has three singular points of multiplicities 4; 2 and 2. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in $\mathbb{P}^3$ with the surprising property that their general projection to $\mathbb{P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of 24 points in $\mathbb{P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.
Wed, 07 Oct 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37992020-10-07T00:00:00ZPokora, PiotrSzemberg, TomaszSzpond, JustynaFelix Klein in course of his study of the regular and its symmetries encountered a highly symmetric configuration of 60 points in $\mathbb{P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the 60 reflection planes in the group $G_{31}$ in the Shephard-Todd list.
In the present note we show that the 60 points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree 6. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of 60 points is a cone with a single singularity of multiplicity 6 and the other has three singular points of multiplicities 4; 2 and 2. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in $\mathbb{P}^3$ with the surprising property that their general projection to $\mathbb{P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of 24 points in $\mathbb{P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property.The Pelletier-Ressayre Hidden Symmetry for Littlewood-Richardson Coefficients
http://publications.mfo.de/handle/mfo/3773
The Pelletier-Ressayre Hidden Symmetry for Littlewood-Richardson Coefficients
Grinberg, Darij
We prove an identity for Littlewood–Richardson coefficients conjectured by Pelletier and Ressayre. The proof relies on a novel birational involution defined over any semifield.
Tue, 08 Sep 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37732020-09-08T00:00:00ZGrinberg, DarijWe prove an identity for Littlewood–Richardson coefficients conjectured by Pelletier and Ressayre. The proof relies on a novel birational involution defined over any semifield.Braidoids
http://publications.mfo.de/handle/mfo/3771
Braidoids
Gügümcü, Neslihan; Lambropoulou, Sofia
Braidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce the notion of braidoids in $\mathbb{R}^2$, a closure operation for braidoids, we prove an analogue of the Alexander theorem, namely an algorithm that turns a knotoid into a braidoid, and we formulate and prove a geometric analogue of the Markov theorem for braidoids using the $L$-moves.
Thu, 03 Sep 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37712020-09-03T00:00:00ZGügümcü, NeslihanLambropoulou, SofiaBraidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce the notion of braidoids in $\mathbb{R}^2$, a closure operation for braidoids, we prove an analogue of the Alexander theorem, namely an algorithm that turns a knotoid into a braidoid, and we formulate and prove a geometric analogue of the Markov theorem for braidoids using the $L$-moves.Maximal Quaternion Orders in Quadratic Extensions - in Hurwitz’s Diaries
http://publications.mfo.de/handle/mfo/3768
Maximal Quaternion Orders in Quadratic Extensions - in Hurwitz’s Diaries
Oswald, Nicola; Steuding, Jörn
We present and comment on some unpublished work of Adolf Hurwitz on quaternion arithmetic from his diaries.
Mon, 03 Aug 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37682020-08-03T00:00:00ZOswald, NicolaSteuding, JörnWe present and comment on some unpublished work of Adolf Hurwitz on quaternion arithmetic from his diaries.