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- ItemMehler-Heine asymptotics of a class of generalized hypergeometric polynomials(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Bracciali, Cleonice F.; Moreno-Balcázar, Juan JoséWe obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures.
- ItemA graphical interface for the Gromov-Witten theory of curves(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Cavalieri, Renzo; Johnson, Paul; Markwig, Hannah; Ranganathan, DhruvWe explore the explicit relationship between the descendant Gromov–Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov–Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient “graphical user interface” for Okounkov and Pandharipande’s celebrated GW/H correspondence. An important goal of this paper is to spell out the connections between these various perspectives for target dimension 1, as a first step in studying the analogous relationship between logarithmic descendant theory, tropical curve counting, and Fock space formalisms in higher dimensions.
- ItemFibonacci-like unimodal inverse limit spaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Bruin, H.; Štimac, S.We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allows us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to simplification of some existing results, including the Ingram Conjecture for Fibonacci-like unimodal inverse limits.
- ItemAnalytic structure in fibers(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Aron, Richard M.; Falcó, Javier; García, Domingo; Maestre, ManuelLet BX be the open unit ball of a complex Banach space X, and let H∞(BX) and Au(BX) be, respectively, the algebra of bounded holomorphic functions on BX and the subalgebra of uniformly continuous holomorphic functions on BX. In this paper we study the analytic structure of fibers in the spectrum of these two algebras. For the case of H∞(BX), we prove that the fiber in M(H∞(Bc0)) over any point of the distinguished boundary of the closed unit ball B¯ℓ∞ of ℓ∞ contains an analytic copy of Bℓ∞. In the case of Au(BX) we prove that if there exists a polynomial whose restriction to the open unit ball of X is not weakly continuous at some point, then the fiber over every point of the open unit ball of the bidual contains an analytic copy of D.
- ItemOn densities of lattice arrangements intersecting every i-dimensional affine subspace(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Merino, Bernardo González; Henze, MatthiasIn 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski’s fundamental theorem, and which is strongly linked to the well-known Mahlerconjecture. Based on the covering minima of Kannan & Lovász and a problem posed by Fejes Tóth, we arrange Makai Jr.’s conjecture into a wider context and investigate densities of lattice arrangements of convex bodies intersecting every i-dimensional affine subspace. Then it becomes natural also to formulate and study a dual analog to Minkowski’s second fundamental theorem. As our main results, we derive meaningful asymptotic lower bounds for the densities of such arrangements, and furthermore, we solve the problems exactly for the special, yet important, class of unconditional convex bodies.
- ItemDominance and transmissions in supertropical valuation theory(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisThis paper is a sequel of [IKR1], where we defined supervaluations on a commutative ring R and studied a dominance relation Φ>=v between supervaluations φ and υ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation φ:R→U is a multiplicative map from R to a supertropical semiring U, cf. [IR1], [IR2], [IKR1], with further properties, which mean that φ is a sort of refinement, or covering, of an m-valuation (= monoid valuation) υ:R→M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [B], while φ>=υ means that υ:R→V is a sort of coarsening of the supervaluation φ. If φ(R) generates the semiring U, then φ>=υ if there exists a "transmission" α:U→V with φ=α∘φ. Transmissions are multiplicative maps with further properties, cf. [IKR1, §55]. Every semiring homomorphism α:U→V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the paper we study surjective transmissions via equivalence relations on supertropical semirings, often much more complicated than congruences by ideals in usual commutative algebra.
- ItemGhost algebras of double Burnside algebras via Schur functors(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Boltje, Robert; Danz, SusanneFor a finite group G, we introduce a multiplication on the Q-vector space with basis SG×G, the set of subgroups of G × G. The resulting Q-algebra A˜ can be considered as a ghost algebra for the double Burnside ring B(G,G) in the sense that the mark homomorphism from B(G,G) to A˜ is a ring homomorphism. Our approach interprets QB(G,G) as an algebra eAe, where A is a twisted monoid algebra and e is an idempotent in A. The monoid underlying the algebra A is again equal to SG×G with multiplication given by composition of relations (when a subgroup of G × G is interpreted as a relation between G and G). The algebras A and A˜ are isomorphic via Mo¨bius inversion in the poset SG×G. As an application we improve results by Bouc on the parametrization of simple modules of QB(G,G) and also of simple biset functors, by using results by Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where G is a cyclic group of order n, we give an explicit isomorphism between QB(G,G) and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order k, where k divides n.
- ItemPolynomiality, wall crossings and tropical geometry of rational double Hurwitz cycles(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Bertram, Aaron; Cavalieri, Renzo; Markwig, HannahWe study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.
- ItemAnalytic varieties with finite volume amoebas are algebraic(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Madani, Farid; Nisse, MounirIn this paper, we study the amoeba volume of a given k-dimensional generic analytic variety V of the complex algebraic torus (C∗)n. When n>=2k, we show that V is algebraic if and only if the volume of its amoeba is finite. Moreover, in this case, we establish a comparison theorem for the volume of the amoeba and the coamoeba. Examples and applications to the k-linear spaces will be given.
- ItemRight unimodal and bimodal singularities in positive characteristic(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Nguyen, Hong-DucThe problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t. right equivalence. The classification of right simple singularities in positive characteristic was achieved by Greuel and the author in 2014. In the present paper we classify right unimodal and bimodal singularities in positive characteristic by giving explicit normal forms. Moreover we completely determine all possible adjacency diagrams of simple, unimodal and bimodal singularities. As an application we prove that, for singularities of right modality at most 2, the µ-constant stratum is smooth and its dimension is equal to the right modality. In contrast to the complex analytic case, there are, for any positive characteristic, only finitely many 1-dimensional (resp. 2-dimensional) families of right class of unimodal (resp. bimodal) singularities. We show that for fixed characteristic p > 0 of the ground field, the Milnor number of f satisfies µ(f) 4p, if the right modality of f is at most 2.