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- ItemSpherical arc-length as a global conformal parameter for analytic curves in the Riemann sphere(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Gauthier, Paul; Nestoridis, Vassili; Papadopoulos, AthanaseWe prove that for every analytic curve in the complex plane C, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Rn and Cn and we discuss the situation of curves in the Riemann sphere C {∞}.
- 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.
- ItemNon-extendability of holomorphic functions with bounded or continuously extendable derivatives(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Moschonas, Dionysios; Nestoridis, VassiliWe consider the spaces H∞F(Ω) and AF(Ω) containing all holomorphic functions f on an open set Ω⊆C, such that all derivatives f(l), l∈F⊆N0={0,1,...}, are bounded on Ω, or continuously extendable on Ω¯¯¯¯, respectively. We endow these spaces with their natural topologies and they become Fr\'echet spaces. We prove that the set S of non-extendable functions in each of these spaces is either void, or dense and Gδ. We give examples where S=∅ or not. Furthermore, we examine cases where F can be replaced by F˜={l∈N0:minF⩽l⩽supF}, or F˜0={l∈N0:0⩽l⩽supF} and the corresponding spaces stay unchanged.
- ItemYet another algorithm for the symmetric eigenvalue problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Aurentz, Jared L.; Mach, Thomas; Vandebril, Raf; Watkins, David S.In this paper we present a new algorithm for solving the symmetric matrix eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg’s implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable and also demonstrate that this algorithm is comparable with other well known implementations in terms of both speed and accuracy.
- ItemThe colored Jones polynomial and Kontsevich-Zagier series for double twist knots(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Lovejoy, Jeremy; Osburn, RobertUsing a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots K(−m,−p) and K(−m,p) where m and p are positive integers. In the (−m,−p) case, this leads to new families of q-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of K(m,p) gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function for strongly unimodal sequences.
- ItemLegendrian lens space surgeries(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Geiges, Hansjörg; Onaran, SinemWe show that every tight contact structure on any of the lens spaces L(ns2 - s + 1; s2) with n ≥ 2, s ≥ 1, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot T(s - (sn - 1)) in the tight or an overtwisted contact structure on the 3-sphere.
- ItemComposition of irreducible morphisms in coils(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Chaio, Claudia; Malicki, PiotrWe study the non-zero composition of n irreducible morphisms between modules lying in coils in relation with the powers of the radical of their module category.
- ItemOn an effective variation of Kronecker’s approximation theorem avoiding algebraic sets(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Fukshansky, Lenny; German, Oleg; Moshchevitin, NikolayLet Λ⊂Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z⊂Rn be the zero locus of a finite collection of polynomials such that Λ⊈Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1,…,Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε>0 and a∈Rn, we prove the existence of a vector x∈Λ∖Z of explicitly bounded sup-norm such that ∥Li(x)−ai∥<ε for each 1≤i≤t, where ∥ ∥ stands for the distance to the nearest integer. The bound on sup-norm of x depends on ε, as well as on Λ, K, Z and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of Λ∖Z under the linear forms L1,…,Lt in the t-torus~Rt/Zt. In the appendix, we also discuss a construction of badly approximable matrices, a subject closely related to our proof of effective Kronecker's theorem, via Liouville-type inequalities and algebraic transference principles.
- ItemKilling tensors on tori(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Heil, Konstantin; Moroianu, Andrei; Semmelmann, UweWe show that Killing tensors on conformally at n-dimensional tori whose con- formal factor only depends on one variable, are polynomials in the metric and in the Killing vector elds. In other words, every rst integral of the geodesic ow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta.
- 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.