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- ItemLocalized endomorphisms of graph algebras(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Conti, Roberto; Hong, Jeong Hee; Szyma´nski, WojciechEndomorphisms of graph C*-algebras are investigated. A combinatorial ap- proach to analysis of permutative endomorphisms is developed. Then invertibility criteria for localized endomorphisms are given. Furthermore, proper endomor- phisms which restrict to automorphisms of the canonical diagonal MASA are analyzed. The Weyl group and the restricted Weyl group of a graph C*-algebra are introduced and investigated. Criteria of outerness for automorphisms in the restricted Weyl group are found.
- ItemOn reflection subgroups of finite Coxeter groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, GerhardLet W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.
- ItemBraid equivalences and the L-moves(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Lambropoulou, SofiaIn this survey paper we present the L–moves between braids and how they can adapt and serve for establishing and proving braid equivalence theorems for various diagrammatic settings, such as for classical knots, for knots in knot complements, in c.c.o. 3–manifolds and in handlebodies, as well as for virtual knots, for flat virtuals, for welded knots and for singular knots. The L–moves are local and they provide a uniform ground for formulating and proving braid equivalence theorems for any diagrammatic setting where the notion of braid and diagrammatic isotopy is defined, the statements being first geometric and then algebraic.
- ItemThere is a unique real tight contact 3-ball(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Öztürk, Ferit; Salepciİ, NerminWe prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional direction. For the proofs, we use the real versions of contact neighborhood theorems and the invariant convex surface theory in real contact manifolds.
- 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.
- ItemSupertropical semirings and supervaluations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisWe interpret a valuation v on a ring R as a map v : R ! M into a so called bipotent semiring M (the usual max-plus setting), and then de¯ne a supervaluation ' as a suitable map into a supertropical semiring U with ghost ideal M (cf. [IR1], [IR2]) covering v via the ghost map U ! M. The set Cov(v) of all supervaluations covering v has a natural ordering which makes it a complete lattice. In the case that R is a field, hence for v a Krull valuation, we give a complete explicit description of Cov(v). The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.
- ItemAverages of shifted convolutions of d3(n)(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Baier, S.; Browning, T.D.; Marasingha, G.; Zhao, L.We investigate the first and second moments of shifted convolutions of the generalised divisor function d3(n).
- ItemSemi-invertible extensions of C*-algebras(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Manuilov, Vladimir; Thomsen, KlausWe prolonge the list of C*-algebras for which all extensions by any stable separable C*-algebra are semi-invertible. In particular, we handle certain amalgamations, both of C*-algebras and of groups. Concerning groups we consider both reduced and full group C*-algebras.
- ItemLegendrian knots in Lens spaces(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Onaran, Sinem CelikIn this note, we first classify all topological torus knots lying on the Heegaard torus in Lens spaces, and then we classify Legendrian representatives of torus knots. We show that all Legendrian torus knots in universally tight contact structures on Lens spaces are determined up to contactomorphism by their knot type, rational Thurston-Bennequin invariant and rational rotation number.
- ItemSupertropical linear algebra(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisThe objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of \ghost surpasses." Special attention is paid to the various notions of \base," which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in a d-base may vary according to the d-base, it is shown that when an s-base exists, it is unique up to permutation and multiplication by scalars, and can be identi¯ed with a set of \critical" elements. Linear functionals and the dual space are also studied, leading to supertropical bilinear forms and a supertropical version of the Gram matrix, including its connection to linear dependence, as well as a supertropical version of a theorem of Artin.