Oberwolfach TEST Repository
http://publications.mfo.de:8080
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Tue, 09 Jan 2018 19:18:27 GMT2018-01-09T19:18:27ZOberwolfach TEST Repositoryhttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:8080
Z2-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 $\widetilde{\text{SL}_2(\mathbb{R})}.$
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.
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.A few shades of interpolation
http://publications.mfo.de/handle/mfo/1329
A few shades of interpolation
Szpond, Justyna
The topic of this snapshot is interpolation. In the
ordinary sense, interpolation means to insert something
of a different nature into something else. In
mathematics, interpolation means constructing new
data points from given data points. The new points
usually lie in between the already-known points. The
purpose of this snapshot is to introduce a particular
type of interpolation, namely, polynomial interpolation.
This will be explained starting from basic ideas
that go back to the ancient Babylonians and Greeks,
and will arrive at subjects of current research activity.
Thu, 07 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13292017-12-07T00:00:00ZSzpond, JustynaThe topic of this snapshot is interpolation. In the
ordinary sense, interpolation means to insert something
of a different nature into something else. In
mathematics, interpolation means constructing new
data points from given data points. The new points
usually lie in between the already-known points. The
purpose of this snapshot is to introduce a particular
type of interpolation, namely, polynomial interpolation.
This will be explained starting from basic ideas
that go back to the ancient Babylonians and Greeks,
and will arrive at subjects of current research activity.Closed geodesics on surfaces and Riemannian manifolds
http://publications.mfo.de/handle/mfo/1328
Closed geodesics on surfaces and Riemannian manifolds
Radeschi, Marco
Geodesics are special paths in surfaces and so-called
Riemannian manifolds which connect close points
in the shortest way. Closed geodesics are geodesics
which go back to where they started. In this snapshot
we talk about these special paths, and the efforts to
find closed geodesics.
Thu, 07 Dec 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13282017-12-07T00:00:00ZRadeschi, MarcoGeodesics are special paths in surfaces and so-called
Riemannian manifolds which connect close points
in the shortest way. Closed geodesics are geodesics
which go back to where they started. In this snapshot
we talk about these special paths, and the efforts to
find closed geodesics.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 $\mathcal D$.
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.
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.
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.K-Theory for Group C*-Algebras and Semigroup C*-Algebras
http://publications.mfo.de/handle/mfo/1324
K-Theory for Group C*-Algebras and Semigroup C*-Algebras
Cuntz, Joachim; Echterhoff, Siegfried; Li, Xin; Yu, Guoliang
This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions.
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13242017-01-01T00:00:00ZCuntz, JoachimEchterhoff, SiegfriedLi, XinYu, GuoliangThis book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions.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.
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
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)
Mon, 02 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13222017-10-02T00:00:00ZThe 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.