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.Wed, 17 Aug 2022 07:18:50 GMT2022-08-17T07:18:50ZEmbedding Spaces of Split Links
http://publications.mfo.de/handle/mfo/3966
Embedding Spaces of Split Links
Boyd, Rachael; Bregman, Corey
We study the homotopy type of the space $\mathcal{E}(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space
of separating systems and show that this is homotopy equivalent to $\mathcal{E}(L)$. This combinatorial object provides a gateway to studying the homotopy type of $\mathcal{E}(L)$ via the homotopy type of the spaces $\mathcal{E}(L_i)$. We apply this tool to find a simple description of the fundamental group, or motion group, of $\mathcal{E}(L)$, and extend this to a description of the motion group of embeddings in $S^3$.
Mon, 01 Aug 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39662022-08-01T00:00:00ZBoyd, RachaelBregman, CoreyWe study the homotopy type of the space $\mathcal{E}(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space
of separating systems and show that this is homotopy equivalent to $\mathcal{E}(L)$. This combinatorial object provides a gateway to studying the homotopy type of $\mathcal{E}(L)$ via the homotopy type of the spaces $\mathcal{E}(L_i)$. We apply this tool to find a simple description of the fundamental group, or motion group, of $\mathcal{E}(L)$, and extend this to a description of the motion group of embeddings in $S^3$.Shock-avoiding Slicing Conditions: Tests and Calibrations
http://publications.mfo.de/handle/mfo/3963
Shock-avoiding Slicing Conditions: Tests and Calibrations
Baumgarte, Thomas W.; Hilditch, David
While the 1+log slicing condition has been extremely successful in numerous numerical relativity simulations, it is also known to develop "gauge-shocks" in some examples. Alternative "shockavoiding" slicing conditions suggested by Alcubierre prevent these pathologies in those examples, but have not yet been explored and tested very broadly. In this paper we compare the performance of shock-avoiding slicing conditions with those of 1+log slicing for a number of "text-book" problems, including black holes and relativistic stars. While, in some simulations, the shock-avoiding slicing conditions feature some unusual properties and lead to more "gauge-dynamics" than the 1+log slicing condition, we find that they perform quite similarly in terms of stability and accuracy, and hence provide a very viable alternative to 1+log slicing.
Tue, 19 Jul 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39632022-07-19T00:00:00ZBaumgarte, Thomas W.Hilditch, DavidWhile the 1+log slicing condition has been extremely successful in numerous numerical relativity simulations, it is also known to develop "gauge-shocks" in some examples. Alternative "shockavoiding" slicing conditions suggested by Alcubierre prevent these pathologies in those examples, but have not yet been explored and tested very broadly. In this paper we compare the performance of shock-avoiding slicing conditions with those of 1+log slicing for a number of "text-book" problems, including black holes and relativistic stars. While, in some simulations, the shock-avoiding slicing conditions feature some unusual properties and lead to more "gauge-dynamics" than the 1+log slicing condition, we find that they perform quite similarly in terms of stability and accuracy, and hence provide a very viable alternative to 1+log slicing.On the Enumeration of Finite $L$-Algebras
http://publications.mfo.de/handle/mfo/3961
On the Enumeration of Finite $L$-Algebras
Dietzel, Carsten; Menchón, Paula; Vendramin, Leandro
We use Constraint Satisfaction Methods to construct and enumerate finite L-algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang-Baxter equation. There are 377322225 isomorphism classes of $L$-algebras of size eight. The database constructed suggest the existence of bijections between certain classes of $L$-algebras and well-known combinatorial objects. On the one hand, we prove that Bell numbers enumerate isomorphism classes of finite linear $L$-algebras. On the other hand, we also prove that finite regular $L$-algebras are in bijective correspondence with infinite-dimensional Young diagrams.
Wed, 29 Jun 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39612022-06-29T00:00:00ZDietzel, CarstenMenchón, PaulaVendramin, LeandroWe use Constraint Satisfaction Methods to construct and enumerate finite L-algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang-Baxter equation. There are 377322225 isomorphism classes of $L$-algebras of size eight. The database constructed suggest the existence of bijections between certain classes of $L$-algebras and well-known combinatorial objects. On the one hand, we prove that Bell numbers enumerate isomorphism classes of finite linear $L$-algebras. On the other hand, we also prove that finite regular $L$-algebras are in bijective correspondence with infinite-dimensional Young diagrams.Discretization of Inherent ODEs and the Geometric Integration of DAEs with Symmetries
http://publications.mfo.de/handle/mfo/3953
Discretization of Inherent ODEs and the Geometric Integration of DAEs with Symmetries
Kunkel, Peter; Mehrmann, Volker
Discretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.
Wed, 08 Jun 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39532022-06-08T00:00:00ZKunkel, PeterMehrmann, VolkerDiscretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.Bounded Weight Modules for Basic Classical Lie Superalgebras at Infinity
http://publications.mfo.de/handle/mfo/3951
Bounded Weight Modules for Basic Classical Lie Superalgebras at Infinity
Grantcharov, Dimitar; Penkov, Ivan; Serganova, Vera
We classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modules and oscillator-type $\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\mathfrak{osp} (m | 2n)$ (under the assumption $\dim \, \mathfrak{osp} (m | 2n) = \infty$) by reducing its study to already
known categories of representations of $\mathfrak{sp} (2n)$, where $n$ possibly
equals $\infty$. When classifying simple bounded weight $\mathfrak{sl}(\infty |\infty)$-modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra $\mathfrak{sl}(\infty |\infty)_{\bar{0}}$. We finish the paper by establishing some first facts about the category of bounded weight $\mathfrak{sl} (\infty |\infty)$-modules.
Mon, 30 May 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39512022-05-30T00:00:00ZGrantcharov, DimitarPenkov, IvanSerganova, VeraWe classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modules and oscillator-type $\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\mathfrak{osp} (m | 2n)$ (under the assumption $\dim \, \mathfrak{osp} (m | 2n) = \infty$) by reducing its study to already
known categories of representations of $\mathfrak{sp} (2n)$, where $n$ possibly
equals $\infty$. When classifying simple bounded weight $\mathfrak{sl}(\infty |\infty)$-modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra $\mathfrak{sl}(\infty |\infty)_{\bar{0}}$. We finish the paper by establishing some first facts about the category of bounded weight $\mathfrak{sl} (\infty |\infty)$-modules.Coorbit Spaces and Dual Molecules: the Quasi-Banach Case
http://publications.mfo.de/handle/mfo/3950
Coorbit Spaces and Dual Molecules: the Quasi-Banach Case
Van Velthoven, Jordy Timo; Voigtlaender, Felix
This paper provides a self-contained exposition of coorbit spaces associated with integrable group representations and quasi-Banach function spaces. It extends the theory in [Studia Math., 180(3):237–253, 2007] to locally compact groups that do not necessarily possess a compact, conjugation-invariant unit neighborhood. Furthermore, the present paper establishes the existence of dual molecules of frames and Riesz sequences as in [J. Funct. Anal., 280(10):56, 2021] for the setting of quasi-Banach spaces. To ensure the direct applicability to various well-studied examples, the theory is developed for possibly projective and reducible unitary representations.
Fri, 27 May 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39502022-05-27T00:00:00ZVan Velthoven, Jordy TimoVoigtlaender, FelixThis paper provides a self-contained exposition of coorbit spaces associated with integrable group representations and quasi-Banach function spaces. It extends the theory in [Studia Math., 180(3):237–253, 2007] to locally compact groups that do not necessarily possess a compact, conjugation-invariant unit neighborhood. Furthermore, the present paper establishes the existence of dual molecules of frames and Riesz sequences as in [J. Funct. Anal., 280(10):56, 2021] for the setting of quasi-Banach spaces. To ensure the direct applicability to various well-studied examples, the theory is developed for possibly projective and reducible unitary representations.Deciding Non-Freeness of Rational Möbius Groups
http://publications.mfo.de/handle/mfo/3929
Deciding Non-Freeness of Rational Möbius Groups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
We explore a new computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, $\mathbb{Q}$). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, $R$) for a localization $R$ = $\mathbb{Z}$[$\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a Möbius subgroup $G$ is not free by showing that it has finite index in the relevant SL(2, $R$). Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in SL(2, $R$) that contains $G$.
Tue, 22 Mar 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39292022-03-22T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWe explore a new computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, $\mathbb{Q}$). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, $R$) for a localization $R$ = $\mathbb{Z}$[$\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a Möbius subgroup $G$ is not free by showing that it has finite index in the relevant SL(2, $R$). Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in SL(2, $R$) that contains $G$.Characterization of Tropical Planar Curves up to Genus Six
http://publications.mfo.de/handle/mfo/3928
Characterization of Tropical Planar Curves up to Genus Six
Tewari, Ayush Kumar
We provide new forbidden criterion for realizability of smooth tropical plane curves. This in turn provides us a complete classification of smooth tropical plane curves up to genus six.
Wed, 16 Mar 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39282022-03-16T00:00:00ZTewari, Ayush KumarWe provide new forbidden criterion for realizability of smooth tropical plane curves. This in turn provides us a complete classification of smooth tropical plane curves up to genus six.Local and Global Canonical Forms for Differential-Algebraic Equations with Symmetries
http://publications.mfo.de/handle/mfo/3921
Local and Global Canonical Forms for Differential-Algebraic Equations with Symmetries
Kunkel, Peter; Mehrmann, Volker
Linear time-varying differential-algebraic equations with symmetries are studied. The structures that we address are self-adjoint and skew-adjoint systems. Local and global canonical forms under congruence are presented and used to classify the geometric properties of the flow associated with the differential equation as symplectic or generalized orthogonal flow. As applications, the results are applied to the analysis of dissipative Hamiltonian
systems arising from circuit simulation and incompressible flow.
Mon, 21 Feb 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39212022-02-21T00:00:00ZKunkel, PeterMehrmann, VolkerLinear time-varying differential-algebraic equations with symmetries are studied. The structures that we address are self-adjoint and skew-adjoint systems. Local and global canonical forms under congruence are presented and used to classify the geometric properties of the flow associated with the differential equation as symplectic or generalized orthogonal flow. As applications, the results are applied to the analysis of dissipative Hamiltonian
systems arising from circuit simulation and incompressible flow.Aeppli-Bott-Chern-Massey Products, Bigraded Notions of Formality, and Non-Zero Degree Maps
http://publications.mfo.de/handle/mfo/3920
Aeppli-Bott-Chern-Massey Products, Bigraded Notions of Formality, and Non-Zero Degree Maps
Milivojević, Aleksandar; Stelzig, Jonas
We introduce and study notions of bigraded formality for the algebra of forms on a complex manifold, along with their relation to higher Aeppli-Bott-Chern-Massey products which extend, in an augmented setting, the case of triple products studied by Angella-Tomassini. We show that these Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to the blow-up along a complex submanifold, as long their degree is less than the real codimension of the submanifold. We then consider the general question of under which conditions formality is preserved by non-zero degree maps.
Tue, 15 Feb 2022 00:00:00 GMThttp://publications.mfo.de/handle/mfo/39202022-02-15T00:00:00ZMilivojević, AleksandarStelzig, JonasWe introduce and study notions of bigraded formality for the algebra of forms on a complex manifold, along with their relation to higher Aeppli-Bott-Chern-Massey products which extend, in an augmented setting, the case of triple products studied by Angella-Tomassini. We show that these Aeppli-Bott-Chern-Massey products on complex manifolds pull back non-trivially to the blow-up along a complex submanifold, as long their degree is less than the real codimension of the submanifold. We then consider the general question of under which conditions formality is preserved by non-zero degree maps.