2017
http://publications.mfo.de/handle/mfo/1349
Fri, 01 Dec 2023 04:48:02 GMT2023-12-01T04:48:02ZZ2-Thurston Norm and Complexity of 3-Manifolds, II
http://publications.mfo.de/handle/mfo/1331
Z2-Thurston Norm and Complexity of 3-Manifolds, II
Jaco, William; Rubinstein, J. Hyam; Spreer, Jonathan; Tillmann, Stephan
In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3-manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry $\widetilde{\text{SL}_2(\mathbb{R})}.$
Research in Pairs 2017
Wed, 20 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13312017-12-20T00:00:00ZJaco, WilliamRubinstein, J. HyamSpreer, JonathanTillmann, StephanIn this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3-manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry $\widetilde{\text{SL}_2(\mathbb{R})}.$Gradient Canyons, Concentration of Curvature, and Lipschitz Invariants
http://publications.mfo.de/handle/mfo/1330
Gradient Canyons, Concentration of Curvature, and Lipschitz Invariants
Paunescu, Laurentiu; Tibăr, Mihai-Marius
We find new bi-Lipschitz invariants of holomorphic functions of two variables by using the gradient canyons and by combining analytic and geometric viewpoints on the concentration of curvature.
Research in Pairs 2017
Wed, 13 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13302017-12-13T00:00:00ZPaunescu, LaurentiuTibăr, Mihai-MariusWe find new bi-Lipschitz invariants of holomorphic functions of two variables by using the gradient canyons and by combining analytic and geometric viewpoints on the concentration of curvature.Bredon Cohomology and Robot Motion Planning
http://publications.mfo.de/handle/mfo/1327
Bredon Cohomology and Robot Motion Planning
Farber, Michael; Grant, Mark; Lupton, Gregory; Oprea, John
In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$ can be characterised as the smallest integer $k$ such that the canonical $\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol $E(\pi\times\pi)$ denotes the classifying space for free actions and $E_{\mathcal D}(\pi times\pi)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3, {\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.
Research in Pairs 2017
Wed, 29 Nov 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13272017-11-29T00:00:00ZFarber, MichaelGrant, MarkLupton, GregoryOprea, JohnIn this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$ can be characterised as the smallest integer $k$ such that the canonical $\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol $E(\pi\times\pi)$ denotes the classifying space for free actions and $E_{\mathcal D}(\pi times\pi)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3, {\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.Experimenting with Zariski Dense Subgroups
http://publications.mfo.de/handle/mfo/1326
Experimenting with Zariski Dense Subgroups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
We give a method to describe all congruence images of a finitely generated Zariski dense group $H\leq SL(n, \mathbb{R})$. The method is applied to obtain efficient algorithms for solving this problem in odd prime degree $n$; if $n=2$ then we compute all congruence images only modulo primes. We propose a separate method that works for all $n$ as long as $H$ contains a known transvection. The algorithms have been implemented in ${\sf GAP}$, enabling computer experiments with important classes of linear groups that have recently emerged.
Research in Pairs 2017
Sat, 28 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13262017-10-28T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWe give a method to describe all congruence images of a finitely generated Zariski dense group $H\leq SL(n, \mathbb{R})$. The method is applied to obtain efficient algorithms for solving this problem in odd prime degree $n$; if $n=2$ then we compute all congruence images only modulo primes. We propose a separate method that works for all $n$ as long as $H$ contains a known transvection. The algorithms have been implemented in ${\sf GAP}$, enabling computer experiments with important classes of linear groups that have recently emerged.The Varchenko Determinant of a Coxeter Arrangement
http://publications.mfo.de/handle/mfo/1325
The Varchenko Determinant of a Coxeter Arrangement
Pfeiffer, Götz; Randriamaro, Hery
The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.
OWLF 2017
Fri, 24 Nov 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13252017-11-24T00:00:00ZPfeiffer, GötzRandriamaro, HeryThe Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.Looking Back on Inverse Scattering Theory
http://publications.mfo.de/handle/mfo/1323
Looking Back on Inverse Scattering Theory
Colton, David; Kress, Rainer
We present an essay on the mathematical development of inverse scattering theory for time-harmonic waves during the past fifty years together with some personal memories of our participation in these
events.
Research in Pairs 2017
Thu, 05 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13232017-10-05T00:00:00ZColton, DavidKress, RainerWe present an essay on the mathematical development of inverse scattering theory for time-harmonic waves during the past fifty years together with some personal memories of our participation in these
events.Geometry of Free Loci and Factorization of Noncommutative Polynomials
http://publications.mfo.de/handle/mfo/1322
Geometry of Free Loci and Factorization of Noncommutative Polynomials
Helton, J. William; Klep, Igor; Volčič, Jurij; Helton, J. William
The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.
MSC 2010: 13J30; 15A22; 47A56 (Primary) | 14P10; 16U30; 16R30 (Secondary); Research in Pairs 2017
Mon, 02 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13222017-10-02T00:00:00ZHelton, J. WilliamKlep, IgorVolčič, JurijHelton, J. WilliamThe free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.GAP Functionality for Zariski Dense Groups
http://publications.mfo.de/handle/mfo/1321
GAP Functionality for Zariski Dense Groups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
In this document we describe the functionality of GAP [4] routines for Zariski dense or arithmetic groups that are developed in [1, 2, 3]. The research underlying the software was supported through the programme "Research in Pairs", at the Mathematisches Forschungsinstitut Oberwolfach, and part of the software was written during a stay in June 2017. The hospitality we received has been greatly appreciated. Our research was also supported by a Marie Sklodowska-Curie Individual Fellowship grant under Horizon 2020 (EU Framework Programme for Research and Innovation), and a Simons Foundation Collaboration Grant Nr. 244502. All support is acknowledged with gratitude.
Research in Pairs 2017
Thu, 14 Sep 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13212017-09-14T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderIn this document we describe the functionality of GAP [4] routines for Zariski dense or arithmetic groups that are developed in [1, 2, 3]. The research underlying the software was supported through the programme "Research in Pairs", at the Mathematisches Forschungsinstitut Oberwolfach, and part of the software was written during a stay in June 2017. The hospitality we received has been greatly appreciated. Our research was also supported by a Marie Sklodowska-Curie Individual Fellowship grant under Horizon 2020 (EU Framework Programme for Research and Innovation), and a Simons Foundation Collaboration Grant Nr. 244502. All support is acknowledged with gratitude.Composition of Irreducible Morphisms in Coils
http://publications.mfo.de/handle/mfo/1320
Composition of Irreducible Morphisms in Coils
Chaio, Claudia; Malicki, Piotr
We study the non-zero composition of n irreducible morphisms between modules lying in coils in relation with the powers of the radical of their module category.
Research in Pairs 2017
Mon, 30 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13202017-10-30T00:00:00ZChaio, ClaudiaMalicki, PiotrWe study the non-zero composition of n irreducible morphisms between modules lying in coils in relation with the powers of the radical of their module category.Non-Extendability of Holomorphic Functions with Bounded or Continuously Extendable Derivatives
http://publications.mfo.de/handle/mfo/1318
Non-Extendability of Holomorphic Functions with Bounded or Continuously Extendable Derivatives
Moschonas, Dionysios; Nestoridis, Vassili
We consider the spaces $H_{F}^{\infty}(\Omega)$ and $\mathcal{A}_{F}(\Omega)$ containing all holomorphic functions $f$ on an open set $\Omega \subseteq \mathbb{C}$, such that all derivatives $f^{(l)}$, $l\in F \subseteq \mathbb{N}_0=\{ 0,1,...\}$, are bounded on $\Omega$, or continuously extendable on $\overline{\Omega}$, respectively. We endow these spaces with their natural topologies and they become Fr\'echet spaces. We prove that the set $S$ of non-extendable functions in each of these spaces is either void, or dense and $G_\delta$. We give examples where $S=\varnothing$ or not. Furthermore, we examine cases where $F$ can be replaced by $\widetilde{F}=\{ l\in \mathbb{N}_0:\min F \leqslant l \leqslant \sup F\}$, or $\widetilde{F}_0= \{l\in \mathbb{N}_0:0\leqslant l \leqslant \sup F\}$ and the corresponding spaces stay unchanged.
Research in Pairs 2016
Sat, 21 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13182017-10-21T00:00:00ZMoschonas, DionysiosNestoridis, VassiliWe consider the spaces $H_{F}^{\infty}(\Omega)$ and $\mathcal{A}_{F}(\Omega)$ containing all holomorphic functions $f$ on an open set $\Omega \subseteq \mathbb{C}$, such that all derivatives $f^{(l)}$, $l\in F \subseteq \mathbb{N}_0=\{ 0,1,...\}$, are bounded on $\Omega$, or continuously extendable on $\overline{\Omega}$, respectively. We endow these spaces with their natural topologies and they become Fr\'echet spaces. We prove that the set $S$ of non-extendable functions in each of these spaces is either void, or dense and $G_\delta$. We give examples where $S=\varnothing$ or not. Furthermore, we examine cases where $F$ can be replaced by $\widetilde{F}=\{ l\in \mathbb{N}_0:\min F \leqslant l \leqslant \sup F\}$, or $\widetilde{F}_0= \{l\in \mathbb{N}_0:0\leqslant l \leqslant \sup F\}$ and the corresponding spaces stay unchanged.