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http://publications.mfo.de:8080
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 23 Sep 2017 20:18:24 GMT2017-09-23T20:18:24ZOberwolfach TEST Repositoryhttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:8080
Aperiodic Order and Spectral Properties
http://publications.mfo.de/handle/mfo/1310
Aperiodic Order and Spectral Properties
Baake, Michael; Damanik, David; Grimm, Uwe
Periodic structures like a typical tiled kitchen floor
or the arrangement of carbon atoms in a diamond
crystal certainly possess a high degree of order. But
what is order without periodicity? In this snapshot,
we are going to explore highly ordered structures that
are substantially nonperiodic, or aperiodic. As we
construct such structures, we will discover surprising
connections to various branches of mathematics, materials
science, and physics. Let us catch a glimpse
into the inherent beauty of aperiodic order!
Thu, 14 Sep 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13102017-09-14T00:00:00ZBaake, MichaelDamanik, DavidGrimm, UwePeriodic structures like a typical tiled kitchen floor
or the arrangement of carbon atoms in a diamond
crystal certainly possess a high degree of order. But
what is order without periodicity? In this snapshot,
we are going to explore highly ordered structures that
are substantially nonperiodic, or aperiodic. As we
construct such structures, we will discover surprising
connections to various branches of mathematics, materials
science, and physics. Let us catch a glimpse
into the inherent beauty of aperiodic order!The Minimal Resolution Conjecture on a general quartic surface in $\mathbb P^3$
http://publications.mfo.de/handle/mfo/1309
The Minimal Resolution Conjecture on a general quartic surface in $\mathbb P^3$
Boij, Mats; Migliore, Juan; Miró-Roig, Rosa M.; Nagel, Uwe
Mustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in $\mathbb P^3$ this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.
MSC: 13D02; 13C40; 13D40; 13E10; 14M06
Thu, 27 Jul 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13092017-07-27T00:00:00ZBoij, MatsMigliore, JuanMiró-Roig, Rosa M.Nagel, UweMustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in $\mathbb P^3$ this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, but now the dimension count does not force the existence of our links.Reducing sub-modules of the Bergman module $\mathbb A^{(\lambda)}(\mathbb D^n)$ under the action of the symmetric group
http://publications.mfo.de/handle/mfo/1308
Reducing sub-modules of the Bergman module $\mathbb A^{(\lambda)}(\mathbb D^n)$ under the action of the symmetric group
Biswas, Shibananda; Ghosh, Gargi; Misra, Gadadhar; Roy, Subrata Shyam
The weighted Bergman spaces on the polydisc, $\mathbb A^{(\lambda)}(\mathbb D^n)$, $\lambda>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ indexed by the
partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $\chi_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $\chi_{\boldsymbol p}(1) \ne \chi_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol
p}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent.
MSC: 47A13; 47B32; 20B30
Thu, 20 Jul 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13082017-07-20T00:00:00ZBiswas, ShibanandaGhosh, GargiMisra, GadadharRoy, Subrata ShyamThe weighted Bergman spaces on the polydisc, $\mathbb A^{(\lambda)}(\mathbb D^n)$, $\lambda>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ indexed by the
partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $\chi_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $\chi_{\boldsymbol p}(1) \ne \chi_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol
p}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(\lambda)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent.An Extension Problem and Trace Hardy Inequality for the Sublaplacian on H-Type Groups
http://publications.mfo.de/handle/mfo/1307
An Extension Problem and Trace Hardy Inequality for the Sublaplacian on H-Type Groups
Roncal, Luz; Thangavelu, Sundaram
In this paper we study the extension problem for the sublaplacian on a H-type group and use the solutions to prove trace Hardy and Hardy inequalities for fractional powers of the sublaplacian.
MSC: 35R03; 22E25; 22E46; 35C15; 35J25
Mon, 24 Jul 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13072017-07-24T00:00:00ZRoncal, LuzThangavelu, SundaramIn this paper we study the extension problem for the sublaplacian on a H-type group and use the solutions to prove trace Hardy and Hardy inequalities for fractional powers of the sublaplacian.Counting Curves on Toric Surfaces Tropical Geometry & the Fock Space
http://publications.mfo.de/handle/mfo/1306
Counting Curves on Toric Surfaces Tropical Geometry & the Fock Space
Cavalieri, Renzo; Johnson, Paul D.; Markwig, Hannah; Ranganathan, Dhruv
We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are shown to coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttche, Cooper–Pandharipande, and Block–Gathmann–Markwig.
Mon, 17 Jul 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13062017-07-17T00:00:00ZCavalieri, RenzoJohnson, Paul D.Markwig, HannahRanganathan, DhruvWe study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are shown to coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttche, Cooper–Pandharipande, and Block–Gathmann–Markwig.The Pseudo-Hyperresolution and Applications
http://publications.mfo.de/handle/mfo/1305
The Pseudo-Hyperresolution and Applications
Nguyen, The Cuong
Resolving objects in an abelian category by injective (projective) resolutions is a fundamental problem in mathematics, and this article aims at introducing a particular solution called “Pseudo-hyperresolutions”. This method originates in the category of unstable modules to study the minimal resolution of the reduced singular cohomology of spheres. In particular, for all integers $n \geq 0$, we can describe a large range of the minimal injective resolution of the sphere $S^n$ based on the Bockstein operation of the Steenrod algebra. Moreover, many classical constructions in algebraic topology, such as the algebraic EHP sequence or the Lambda algebra can be recovered using the Pseudo-hyperresolution method. A particular connection between spheres and the projective spaces is also established. Despite its origin, Pseudo-hyperresolutions generalize to all abelian categories. In particular, many classical construction of injective resolutions of strict polynomial functors can be reunified in view of Pseudo-hyperresolutions. As a consequence, we recover the global dimension of the category of homogeneous strict polynomial functors of finite degree as well as the Mac Lane cohomology of finite fields.
Tue, 04 Jul 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13052017-07-04T00:00:00ZNguyen, The CuongResolving objects in an abelian category by injective (projective) resolutions is a fundamental problem in mathematics, and this article aims at introducing a particular solution called “Pseudo-hyperresolutions”. This method originates in the category of unstable modules to study the minimal resolution of the reduced singular cohomology of spheres. In particular, for all integers $n \geq 0$, we can describe a large range of the minimal injective resolution of the sphere $S^n$ based on the Bockstein operation of the Steenrod algebra. Moreover, many classical constructions in algebraic topology, such as the algebraic EHP sequence or the Lambda algebra can be recovered using the Pseudo-hyperresolution method. A particular connection between spheres and the projective spaces is also established. Despite its origin, Pseudo-hyperresolutions generalize to all abelian categories. In particular, many classical construction of injective resolutions of strict polynomial functors can be reunified in view of Pseudo-hyperresolutions. As a consequence, we recover the global dimension of the category of homogeneous strict polynomial functors of finite degree as well as the Mac Lane cohomology of finite fields.Matrix Elements of Irreducible Representations of SU(n+1) x SU(n+1) and Multivariable Matrix-Valued Orthogonal Polynomials
http://publications.mfo.de/handle/mfo/1304
Matrix Elements of Irreducible Representations of SU(n+1) x SU(n+1) and Multivariable Matrix-Valued Orthogonal Polynomials
Koelink, Erik; van Pruijssen, Maarten; Román, Pablo Manuel
In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case SU(n+1) meets all the conditions that we impose in Part 1. For any kEN0 we obtain families of orthogonal polynomials in n variables with values in theNxN-matrices, where N=((n+k)/k). The case k=0 leads to the classical Heckman-Opdam polynomials of type An with geometric parameter. For k=1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n>=2. We also give explicit expressions of the spherical functions that determine the matrix weight for k = 1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n = 1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n; k) equal to (2; 1) and (3; 1).
Wed, 14 Jun 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13042017-06-14T00:00:00ZKoelink, Erikvan Pruijssen, MaartenRomán, Pablo ManuelIn Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case SU(n+1) meets all the conditions that we impose in Part 1. For any kEN0 we obtain families of orthogonal polynomials in n variables with values in theNxN-matrices, where N=((n+k)/k). The case k=0 leads to the classical Heckman-Opdam polynomials of type An with geometric parameter. For k=1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n>=2. We also give explicit expressions of the spherical functions that determine the matrix weight for k = 1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n = 1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n; k) equal to (2; 1) and (3; 1).Linear Syzygies, Hyperbolic Coxeter Groups and Regularity
http://publications.mfo.de/handle/mfo/1303
Linear Syzygies, Hyperbolic Coxeter Groups and Regularity
Constantinescu, Alexandru; Kahle, Thomas; Varbaro, Matteo
We build a new bridge between geometric group theory and commutative algebra by showing that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection and techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.
MSC: 13F55; 20F55; 13D02
Wed, 24 May 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13032017-05-24T00:00:00ZConstantinescu, AlexandruKahle, ThomasVarbaro, MatteoWe build a new bridge between geometric group theory and commutative algebra by showing that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection and techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.Test 2017
http://publications.mfo.de/handle/mfo/1302
Test 2017
Test
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13022017-01-01T00:00:00ZTestOverlap Synchronisation in Multipartite Random Energy Models
http://publications.mfo.de/handle/mfo/1298
Overlap Synchronisation in Multipartite Random Energy Models
Genovese, Guiseppe; Tantari, Daniele
In a multipartite random energy model, made of coupled GREMs, we determine the joint law of the overlaps in terms of the ones of the single GREMs. This provides the simplest example of the so-called synchronisation of the overlaps.
Sat, 29 Apr 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/12982017-04-29T00:00:00ZGenovese, GuiseppeTantari, DanieleIn a multipartite random energy model, made of coupled GREMs, we determine the joint law of the overlaps in terms of the ones of the single GREMs. This provides the simplest example of the so-called synchronisation of the overlaps.