Proceedings of the London Mathematical Society - current issue

Principal non-commutative torus bundles

Echterhoff, S., Nest, R., Oyono-Oyono, H. — 2009-06-11

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally RKK-trivial. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to Tⁿ-equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal Tⁿ-bundles with H-flux, as studied by Mathai and Rosenberg which possess ‘classical’ T-duals.

On the role of convexity in functional and isoperimetric inequalities

Milman, E. — 2009-06-11

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz’ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, q-log-Sobolev inequalities (q [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ) curvature–dimension condition of Bakry–Émery.

Transversality in families of mappings

Wall, C. T. C. — 2009-06-11

If N and P are smooth manifolds, and Q is a smooth submanifold of P, the set of maps of N to P transverse to Q is residual. However given a family F of smooth mappings, we cannot expect to be able to deform the family to make each member of the family transverse to Q. The development of generic transversality conditions runs in parallel with that of stability theory. We develop a convenient notation, and establish equivalence of different transversality conditions. The first main result expresses equivalence between F inducing a versal unfolding of any multi-germ, multi-transversality of F to contact orbits relative to projection on the parameter space U, and local stability of the projection to U of the pre-image of Q. For global stability we need Q to be closed, the deformation F to be proper, and Q or N compact.

Corresponding results also hold for topological stability in the Thom–Mather sense, and an appropriate notion of topological versality.

To obtain results with Q a stratified subset of P requires an extensive study of theories of contact equivalence relative to a subset of the target. Here we need an analyticity hypothesis on Q; for deeper study of finite determinacy, we need a holonomic condition; even so, the theory is much more complicated. In general, there are properties corresponding to the former ones, but many are weaker. Our second main result, a direct analogue of the first, is only obtained under a hypotheses that F has property (G) at all points and is transverse to Q.

In the topological case, while some results can be obtained, the failure of direct analogues to some basic tools is a bar to further progress.

A class of noncommutative projective surfaces

Rogalski, D., Stafford, J. T. — 2009-06-11

Let A = _i≥0A_i be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A) = k(X)[t, t^–1; ], where is an automorphism of the integral projective surface X. Then we prove that A can be written as a naïve blowup algebra of a projective surface X birational to X. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in qgr-A will always be in (1-1) correspondence with the closed points of the scheme X.

Representations of Lie superalgebras in prime characteristic I

Wang, W., Zhao, L. — 2009-06-11

We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p > 2. A superalgebra generalization of the celebrated Kac–Weisfeiler conjecture is formulated, which exhibits a mixture of p-power and 2-power divisibilities of dimensions of modules. We establish the conjecture for basic classical Lie superalgebras.

On pseudo-harmonic maps in conformal geometry

Kokarev, G. — 2009-06-11

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We discuss existence theorems and obtain an extension of Siu's strong rigidity to Kähler–Weyl geometry. Other applications include topological obstructions to the existence of Kähler–Weyl structures. For example, we show that no co-compact lattice in SO(1, n), n > 2, can be the fundamental group of a compact Kähler–Weyl manifold of certain type.

Constructing smooth manifolds of loop spaces

Stacey, A. — 2009-06-11

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite-dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists. We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations.

Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

Sela, Z. — 2009-06-11

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion-free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first-order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion-free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent.