2018
http://publications.mfo.de/handle/mfo/1350
Fri, 18 Sep 2020 09:42:10 GMT2020-09-18T09:42:10ZDemailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps (Revised Edition)
http://publications.mfo.de/handle/mfo/3694
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps (Revised Edition)
Javanpeykar, Ariyan; Kamenova, Ljudmila
Demailly's conjecture, which is a consequence of the Green-Griffths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety $Y$ to a projective algebraically hyperbolic variety $X$ that map a fixed closed subvariety of $Y$ onto a fixed closed subvariety of $X$ is finite. As an application, we obtain that Aut($X$) is finite and that every surjective endomorphism of $X$ is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffths-Lang conjecture on hyperbolic projective varieties.
Thu, 23 Jan 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36942020-01-23T00:00:00ZJavanpeykar, AriyanKamenova, LjudmilaDemailly's conjecture, which is a consequence of the Green-Griffths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety $Y$ to a projective algebraically hyperbolic variety $X$ that map a fixed closed subvariety of $Y$ onto a fixed closed subvariety of $X$ is finite. As an application, we obtain that Aut($X$) is finite and that every surjective endomorphism of $X$ is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffths-Lang conjecture on hyperbolic projective varieties.The Tutte Polynomial of Ideal Arrangements
http://publications.mfo.de/handle/mfo/1395
The Tutte Polynomial of Ideal Arrangements
Randriamaro, Hery
The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.
Fri, 21 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13952018-12-21T00:00:00ZRandriamaro, HeryThe Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
http://publications.mfo.de/handle/mfo/1394
Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
Beckus, Siegfried; Bellissard, Jean; De Nittis, Giuseppe
The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.
Mon, 17 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13942018-12-17T00:00:00ZBeckus, SiegfriedBellissard, JeanDe Nittis, GiuseppeThe existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.Sur le Minimum de la Fonction de Brjuno
http://publications.mfo.de/handle/mfo/1393
Sur le Minimum de la Fonction de Brjuno
Balazard, Michel; Martin, Bruno
The Brjuno function attains a strict global minimum at the golden section.
Tue, 11 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13932018-12-11T00:00:00ZBalazard, MichelMartin, BrunoThe Brjuno function attains a strict global minimum at the golden section.Criteria for Algebraicity of Analytic Functions
http://publications.mfo.de/handle/mfo/1392
Criteria for Algebraicity of Analytic Functions
Bochnak, Jacek; Gwoździewicz, Janusz; Kucharz, Wojciech
We consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing trough a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.
Mon, 12 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13922018-11-12T00:00:00ZBochnak, JacekGwoździewicz, JanuszKucharz, WojciechWe consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing trough a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.Global Variants of Hartogs' Theorem
http://publications.mfo.de/handle/mfo/1391
Global Variants of Hartogs' Theorem
Bochnak, Jacek; Kucharz, Wojciech
Hartogs' theorem asserts that a separately holomorphic function, defined on an open subset of $\mathbb{C}^n$, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.
Tue, 06 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13912018-11-06T00:00:00ZBochnak, JacekKucharz, WojciechHartogs' theorem asserts that a separately holomorphic function, defined on an open subset of $\mathbb{C}^n$, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.Real Analyticity is Concentrated in Dimension 2
http://publications.mfo.de/handle/mfo/1390
Real Analyticity is Concentrated in Dimension 2
Bochnak, Jacek; Kucharz, Wojciech
We prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to $2$-dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of $C^n$ is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic.
Mon, 05 Nov 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13902018-11-05T00:00:00ZBochnak, JacekKucharz, WojciechWe prove that a real-valued function on a real analytic manifold is analytic whenever all its restrictions to $2$-dimensional analytic submanifolds are analytic functions. We also obtain analogous results in the framework of Nash manifolds and nonsingular real algebraic sets. These results can be regarded as substitutes in the real case for the classical theorem of Hartogs, asserting that a complex-valued function defined on an open subset of $C^n$ is holomorphic if it is holomorphic with respect to each variable separately. In the proofs we use methods of real algebraic geometry even though the initial problem is purely analytic.Computing Congruence Quotients of Zariski Dense Subgroups
http://publications.mfo.de/handle/mfo/1389
Computing Congruence Quotients of Zariski Dense Subgroups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.
Fri, 26 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13892018-10-26T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWe obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.On the Invariants of the Cohomology of Complements of Coxeter Arrangements
http://publications.mfo.de/handle/mfo/1388
On the Invariants of the Cohomology of Complements of Coxeter Arrangements
Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, Gerhard
We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group W. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the space of W-invariants
in this cohomology ring.
Mon, 22 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13882018-10-22T00:00:00ZDouglass, J. MatthewPfeiffer, GötzRöhrle, GerhardWe refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group W. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the space of W-invariants
in this cohomology ring.Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps
http://publications.mfo.de/handle/mfo/1387
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps
Javanpeykar, Ariyan; Kamenova, Ljudmila
Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffiths-Lang conjecture on hyperbolic projective varieties.
Mon, 08 Oct 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13872018-10-08T00:00:00ZJavanpeykar, AriyanKamenova, LjudmilaDemailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffiths-Lang conjecture on hyperbolic projective varieties.