Biembedding Steiner Triple Systems in Surfaces Using the Bose Construction

A uniform framework is presented for biembedding Steiner triple systems obtained from the Bose construction using a cyclic group of odd order, in both orientable and nonorientable surfaces. Within this framework, in the nonorientable case, a formula is given for the number of isomorphism classes and the particular biembedding of Ducrocq and Sterboul (preprint 18pp., 1978) is identified. In the orientable case, it is shown that the biembedding of Grannell et al. (J Combin Des 6 ( 7 ), 325–336) is, up to isomorphism, the unique biembedding of its type. Automorphism groups of the biembeddings are also given.     

Quantum McKay Correspondence for Disc Invariants of Toric Calabi–Yau 3-orbifolds

We announce a result on quantum McK ay correspondence for disc invariants of outer legs in toric Calabi–Yau 3-orbifolds, and illustrate our method in a special example [C3/Z5(1, 1, 3)].     

Hochschild and cyclic (co)homology of superadditive categories

We define the Hochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.     

Factoriality for the reductive Zassenhaus variety and quantum enveloping algebra

Let U(g)$U(\mathfrak{g})$ be the enveloping algebra of a finite dimensional reductive Lie algebra g$\mathfrak{g}$ over an algebraically closed field of prime characteristic. Let _U?,P(s:)$U_{?,P}(\mathfrak{s}:)$ be the simply connected quantum enveloping algebra at the root of unity ?  , of a complex semi-simple finite dimensional Lie algebra s:$\mathfrak{s}:$. We show, by similar proofs, that the centers of both are factorial. While the first result was established by R. Tange [32] (by different methods), the second one confirms a conjecture in [4]. We also provide a general criterion for the factoriality of the centers of enveloping algebras in prime characteristic.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

Channel capacities via p-summing norms

In this paper we show how the metric theory of tensor products developed by Grothendieck perfectly fits in the study of channel capacities, a central topic in Shannon's information theory. Furthermore, in the last years Shannon's theory has been fully generalized to the quantum setting, and revealed qualitatively new phenomena in comparison. In this paper we consider the classical capacity of quantum channels with restricted assisted entanglement. These capacities include the classical capacity and the unlimited entanglement-assisted classical capacity of a quantum channel. Our approach to restricted capacities is based on tools from functional analysis, and in particular the notion of p  -summing maps going back to Grothendieck's work. Pisier's noncommutative vector-valued _Lp$L_p$ spaces allow us to establish the new connection between functional analysis and information theory in the quantum setting.     

On homotopy rigidity of the functor ΣΩ on co-H-spaces

In this paper we study the homotopy rigidity property of the functors ΣΩ and Ω. Our main result is that both functors are homotopy rigid on simply-connected p-local finite co-H-spaces. The result is obtain by a subtle interplay of homotopy decomposition techniques, modular representation theory and the counting principle.     

Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curvature data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self-conformal sets. Our approach based on the classical Renewal Theorem is simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when suitable volume functions (and curvature functions, respectively) related to the generator are used. We also strengthen known results on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case.     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.