2021
http://publications.mfo.de/handle/mfo/3823
Fri, 09 Aug 2024 00:34:55 GMT2024-08-09T00:34:55ZReflection Positivity and Hankel Operators- the Multiplicity Free Case
http://publications.mfo.de/handle/mfo/3906
Reflection Positivity and Hankel Operators- the Multiplicity Free Case
Adamo, Maria Stella; Neeb, Karl-Hermann; Schober, Jonas
We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples $(G,S,\tau)$, where $G$ is a group, $\tau$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $\tau(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method
consists in using the measure $\mu_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$.
Wed, 15 Dec 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39062021-12-15T00:00:00ZAdamo, Maria StellaNeeb, Karl-HermannSchober, JonasWe analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples $(G,S,\tau)$, where $G$ is a group, $\tau$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $\tau(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method
consists in using the measure $\mu_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$.Fundamental Theorem of Projective Geometry over Semirings
http://publications.mfo.de/handle/mfo/3892
Fundamental Theorem of Projective Geometry over Semirings
Tewari, Ayush Kumar
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry.
Mon, 11 Oct 2021 00:00:00 GMThttp://publications.mfo.de/handle/mfo/38922021-10-11T00:00:00ZTewari, Ayush KumarWe state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry.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.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 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.