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- ItemCocycle superrigidity and group actions on stably finite C*-algebras(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Gardella, Eusebio; Lupini, MartinoLet be a countably innite property (T) group, and let A be UHF-algebra of innite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of on A. The proof consists in assigning, to any second countable abelian pro-p group G, a strongly outer action of on A whose (weak) cocycle conjugacy class completely remembers the group G. The group G is reconstructed from the action via its (weak) 1-cohomology set endowed with a canonical pairing function. The key ingredient in this computation is Popa's cocycle superrigidity theorem for Bernoulli shifts on the hypernite II1 factor. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of on A are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when is only assumed to contain an innite subgroup with relative property (T), and A is a (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebra with at least one trace.
- ItemMesh ratios for best-packing and limits of minimal energy configurations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Bondarenko, A.V.; Hardin, D.P.; Saff, E.B.For N-point best-packing configurations ωN on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ρN,A), which is the quotient of the covering radius of ωN relative to A and the minimum pairwise distance between points in ωN . For best-packing configurations ωN that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ωN,A)≤1 and this bound can be attained even for the sphere. In the particular case when N=5 on S1 with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid ω∗5, that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, γ(ω∗5,S2)=1.
- ItemCentral limit theorems for the radial spanning tree(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Schulte, Matthias; Thäle, ChristophConsider a homogeneous Poisson point process in a compact convex set in d- dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing in- tensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length.
- ItemRational approximation on products of planar domains(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Aron, Richard M.; Gauthier, Paul M.; Maestre, Manuel; Nestoridis, Vassili; Falcó, JavierWe consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous with respect to the product topology and separately holomorphic, where I is an arbitrary set and Ui are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off Ui¯¯¯¯¯ are uniformly dense in A(Ω). This generalizes previous results where Ui=D is the open unit disc in C or Ui¯¯¯¯¯c is connected.
- ItemClassification of totally real elliptic Lefschetz fibrations via necklace diagrams(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Salepci, NerminWe show that totally real elliptic Lefschetz brations that admit a real section are classified by their "real loci" which is nothing but an S1-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a necklacediagram. On the one hand, each necklace diagram corresponds to an isomorphism class of a totally real elliptic Lefschetz fibration that admits a real section, and on the other hand, it refers to a decomposition of the identity into a product of certain matrices in PSL(2,Z). Using an algorithm to find such decompositions, we obtain an explicit list of necklace diagrams associated with certain classes of totally real elliptic Lefschetz fibrations. Moreover, we introduce refinements of necklace diagrams and show that refined necklace diagrams determine uniquely the isomorphism classes of the totally real elliptic Lefschetz fibrations which may not have a real section. By means of necklace diagrams we observe some interesting phenomena underlying special feature of real fibrations.
- ItemA real algebra perspective on multivariate tight wavelet frames(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Charina, Maria; Putinar, Mihai; Scheiderer, Claus; Stöckler, JoachimRecent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) from [33] are interpreted in terms of hermitian sums of squares of certain nongenative trigonometric polynomials and in terms of semi-definite programming. The latter together with the results in [31, 35] answer affirmatively the long stading open question of the existence of such tight wavelet frames in dimesion d = 2; we also provide numerically efficient methods for checking their existence and actual construction in any dimension. We exhibit a class of counterexamples in dimension d = 3 showing that, in general, the UEP property is not sufficient for the existence of tight wavelet frames. On the other hand we provide stronger sufficient conditions for the existence of tight wavelet frames in dimension d ≥ 3 and illustrate our results by several examples.
- ItemMultiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Baldon, Velleda; Boysal, Arzu; Vergne, MichèleUsing Szenes formula for multiple Bernoulli series, we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also multiple zeta values.
- ItemTime and band limiting for matrix valued functions, an example(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Grünbaum, F.A.; Pacharoni, I.; Zurrián, I.The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the rst example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of \time and band limiting" admits a commuting local operator. This is a noncommutative analog of the famous prolate spheroidal wave operator.
- ItemContractive idempotents on locally compact quantum groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2012) Neufang, Matthias; Salmi, Pekka; Skalski, Adam; Spronk, Nicogeneral form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.
- ItemThe index of singular zeros of harmonic mappings of anti-analytic degree one(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Luce, Robert; Sète, OlivierWe study harmonic mappings of the form f(z)=h(z)−z¯¯¯, where h is an analytic function. In particular we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of f, where the Jacobian of f is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on the critical set, where the Jacobian vanishes; such zeros are called singular zeros. Our main result is a characterization of the index of singular zeros, which enables one to determine the index directly from the power series of h.