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.Sun, 29 Mar 2020 03:35:09 GMT2020-03-29T03:35:09ZGenerating Finite Coxeter Groups with Elements of the Same Order
http://publications.mfo.de/handle/mfo/3709
Generating Finite Coxeter Groups with Elements of the Same Order
Hart, Sarah; Kelsey, Veronica; Rowley, Peter
Supposing $G$ is a group and $k$ a natural number, $d_k(G)$ is defined to be the minimal number of elements of $G$ of order $k$ which generate $G$ (setting $d_k(G) = 0$ if $G$ has no such generating sets). This paper investigates $d_k(G)$ when $G$ is a finite Coxeter group either of type $B_n$ or $D_n$ or of exceptional type. Together with Garzoni [3] and Yu [10], this determines $d_k(G)$ for all finite irreducible Coxeter groups $G$ when 2$ \leq k \leq$rank$(G)$ (rank$(G) + 1$ when $G$ is of type $A_n$).
Mon, 16 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37092020-03-16T00:00:00ZHart, SarahKelsey, VeronicaRowley, PeterSupposing $G$ is a group and $k$ a natural number, $d_k(G)$ is defined to be the minimal number of elements of $G$ of order $k$ which generate $G$ (setting $d_k(G) = 0$ if $G$ has no such generating sets). This paper investigates $d_k(G)$ when $G$ is a finite Coxeter group either of type $B_n$ or $D_n$ or of exceptional type. Together with Garzoni [3] and Yu [10], this determines $d_k(G)$ for all finite irreducible Coxeter groups $G$ when 2$ \leq k \leq$rank$(G)$ (rank$(G) + 1$ when $G$ is of type $A_n$).Multivariate Hybrid Orthogonal Functions
http://publications.mfo.de/handle/mfo/3706
Multivariate Hybrid Orthogonal Functions
Bracciali, Cleonice F.; Pérez, Teresa E.
We consider multivariate orthogonal functions satisfying hybrid orthogonality conditions
with respect to a moment functional. This kind of orthogonality means that the product of
functions of different parity order is computed by means of the moment functional, and the product of elements of the same parity order is computed by a modification of the original moment functional.
Results about existence conditions, three term relations with matrix coefficients, a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a method to construct bivariate hybrid
orthogonal functions from univariate orthogonal polynomials and univariate orthogonal
functions is presented. Finally, we give a complete description of a sequence of hybrid orthogonal
functions on the unit disk on $\mathbb{R}^2$, that includes, as particular case, the classical orthogonal polynomials on the disk.
Thu, 12 Mar 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/37062020-03-12T00:00:00ZBracciali, Cleonice F.Pérez, Teresa E.We consider multivariate orthogonal functions satisfying hybrid orthogonality conditions
with respect to a moment functional. This kind of orthogonality means that the product of
functions of different parity order is computed by means of the moment functional, and the product of elements of the same parity order is computed by a modification of the original moment functional.
Results about existence conditions, three term relations with matrix coefficients, a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a method to construct bivariate hybrid
orthogonal functions from univariate orthogonal polynomials and univariate orthogonal
functions is presented. Finally, we give a complete description of a sequence of hybrid orthogonal
functions on the unit disk on $\mathbb{R}^2$, that includes, as particular case, the classical orthogonal polynomials on the disk.Splitting Necklaces, with Constraints
http://publications.mfo.de/handle/mfo/3698
Splitting Necklaces, with Constraints
Jojic, Dusko; Panina, Gaiane; Zivaljevic, Rade
We prove several versions of Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results.
(1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace, provided the number of thieves $r=p^\nu$ is a prime power.
(2) The "binary splitting theorem" claims that if $r=2^d$ and the thieves are associated with the vertices of a d-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that
adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [5]) in the case $r=2^d$.
Tue, 11 Feb 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36982020-02-11T00:00:00ZJojic, DuskoPanina, GaianeZivaljevic, RadeWe prove several versions of Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results.
(1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace, provided the number of thieves $r=p^\nu$ is a prime power.
(2) The "binary splitting theorem" claims that if $r=2^d$ and the thieves are associated with the vertices of a d-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that
adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [5]) in the case $r=2^d$.Nondegenerate Invariant Symmetric Bilinear Forms on Simple Lie Superalgebras in Characteristic 2
http://publications.mfo.de/handle/mfo/3697
Nondegenerate Invariant Symmetric Bilinear Forms on Simple Lie Superalgebras in Characteristic 2
Krutov, Andrey; Lebedev, Alexei; Leites, Dimitry; Shchepochkina, Irina
As is well-known, the dimension of the space of non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the ground field is distinct from 2. We prove that in characteristic 2, the superdimension of the space of NISes can be equal to 0, or 1, or 0|1, or 1|1. This superdimension is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple restricted Lie algebra with a NIS (for examples of such Lie algebras, although mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations with both even and odd parameter of several simple finite-dimensional Lie superalgebras in characteristic 2.
Tue, 04 Feb 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36972020-02-04T00:00:00ZKrutov, AndreyLebedev, AlexeiLeites, DimitryShchepochkina, IrinaAs is well-known, the dimension of the space of non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the ground field is distinct from 2. We prove that in characteristic 2, the superdimension of the space of NISes can be equal to 0, or 1, or 0|1, or 1|1. This superdimension is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple restricted Lie algebra with a NIS (for examples of such Lie algebras, although mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations with both even and odd parameter of several simple finite-dimensional Lie superalgebras in characteristic 2.Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
http://publications.mfo.de/handle/mfo/3696
Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
Díaz García, Fredy; Krutov, Andrey; Ó Buachalla, Réamonn; Somberg, Petr; Strung, Karen R.
Noncommutative Kähler structures were recently introduced by the third author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic $1$-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$. Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over $\mathcal{O}_q(G/L_S)$ can be identified as positive, negative, or flat, and hence we can conclude that each Kähler structure is of Fano type. Moreover, it proves possible to extend the Borel-Weil theorem for $\mathcal{O}_q(G/L_S)$ to a direct noncommutative generalisation of the classical Bott-Borel-Weil theorem for positive line bundles.
Mon, 03 Feb 2020 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36962020-02-03T00:00:00ZDíaz García, FredyKrutov, AndreyÓ Buachalla, RéamonnSomberg, PetrStrung, Karen R.Noncommutative Kähler structures were recently introduced by the third author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic $1$-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$. Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over $\mathcal{O}_q(G/L_S)$ can be identified as positive, negative, or flat, and hence we can conclude that each Kähler structure is of Fano type. Moreover, it proves possible to extend the Borel-Weil theorem for $\mathcal{O}_q(G/L_S)$ to a direct noncommutative generalisation of the classical Bott-Borel-Weil theorem for positive line bundles.Demailly’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.Global Solutions to Stochastic Wave Equations with Superlinear Coefficients
http://publications.mfo.de/handle/mfo/3683
Global Solutions to Stochastic Wave Equations with Superlinear Coefficients
Millet, Annie; Sanz-Solé, Marta
We prove existence and uniqueness of a random field solution $(u(t,x);(t,x)\in [0,T]\times \mathbb{R}^d)$ to a stochastic wave equation in dimensions $d=1,2,3$ with diffusion and drift coefficients of the form $|x| \big(
\ln_+(|x|) \big)^a$ for some $a$>0. The proof relies on a sharp analysis of moment estimates of time and space increments of the corresponding stochastic wave equation with globally Lipschitz coefficients. We give examples of spatially correlated Gaussian driving noises where the results apply.
Wed, 13 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36832019-11-13T00:00:00ZMillet, AnnieSanz-Solé, MartaWe prove existence and uniqueness of a random field solution $(u(t,x);(t,x)\in [0,T]\times \mathbb{R}^d)$ to a stochastic wave equation in dimensions $d=1,2,3$ with diffusion and drift coefficients of the form $|x| \big(
\ln_+(|x|) \big)^a$ for some $a$>0. The proof relies on a sharp analysis of moment estimates of time and space increments of the corresponding stochastic wave equation with globally Lipschitz coefficients. We give examples of spatially correlated Gaussian driving noises where the results apply.Matchings and Squarefree Powers of Edge Ideals
http://publications.mfo.de/handle/mfo/3682
Matchings and Squarefree Powers of Edge Ideals
Erey, Nursel; Herzog, Jürgen; Hibi, Takayuki; Saeedi Madani, Sara
Squarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.
Mon, 11 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36822019-11-11T00:00:00ZErey, NurselHerzog, JürgenHibi, TakayukiSaeedi Madani, SaraSquarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.Reflective Prolate-Spheroidal Operators and the KP/KdV Equations
http://publications.mfo.de/handle/mfo/3680
Reflective Prolate-Spheroidal Operators and the KP/KdV Equations
Casper, W. Riley; Grünbaum, F. A.; Yakimov, Milen; Zurrián, Ignacio Nahuel
Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned
concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr $^{ad}$ of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of KP wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90$°$ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal
properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by Airault-McKean-Moser and Krichever, respectively, in the late 70's. Many novel examples are presented.
Tue, 05 Nov 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/36802019-11-05T00:00:00ZCasper, W. RileyGrünbaum, F. A.Yakimov, MilenZurrián, Ignacio NahuelCommuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned
concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr $^{ad}$ of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of KP wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90$°$ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal
properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by Airault-McKean-Moser and Krichever, respectively, in the late 70's. Many novel examples are presented.Groups with Spanier-Whitehead Duality
http://publications.mfo.de/handle/mfo/2518
Groups with Spanier-Whitehead Duality
Nishikawa, Shintaro; Proietti, Valerio
We introduce the notion of Spanier-Whitehead $K$-duality for a discrete group $G$, defined as duality in the KK-category between two $C*$-algebras which are naturally attached to the group, namely the reduced group $C*$-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our
analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.
Tue, 17 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25182019-09-17T00:00:00ZNishikawa, ShintaroProietti, ValerioWe introduce the notion of Spanier-Whitehead $K$-duality for a discrete group $G$, defined as duality in the KK-category between two $C*$-algebras which are naturally attached to the group, namely the reduced group $C*$-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our
analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.