Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1

We develop intrinsic tools for computing the periodic Hopf cyclic cohomology of Hopf algebras related to transverse symmetry in codimension 1. Besides the Hopf algebra found by Connes and the first author in their work on the local index formula for transversely hypoelliptic operators on foliations, this family includes its ‘Schwarzian’ quotient, on which the Rankin-Cohen universal deformation formula is based, the extended Connes-Kreimer Hopf algebra related to renormalization of divergences in QFT, as well as a series of cyclic coverings of these Hopf algebras, motivated by the treatment of transverse symmetry for non-orientable foliations.The method for calculating their Hopf cyclic cohomology is based on two computational devices, which work in tandem and complement each other: one is a spectral sequence for bicrossed product Hopf algebras and the other a Cartan-type homotopy formula in Hopf cyclic cohomology.     

Localization of Chern–Simons type invariants of Riemannian foliations

We prove an Atiyah–Bott–Berline–Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern–Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the union of closed leaves. Various examples are given to illustrate our method.     

Classifying foliations of 3-manifolds via branched surfaces

We use branched surfaces to define an equivalence relation on C¹ codimension one foliations of any closed orientable 3-manifold that are transverse to some fixed nonsingular flow. There is a discrete metric on the set of equivalence classes with the property that foliations that are sufficiently close (up to equivalence) share important topological properties.     

Branched spines and contact structures on 3-manifolds

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in a given homotopy class of plane fields. Entrata in Redazione il 6 novembre 1998.     

Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras

We define the notion of Connes-von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern-Connes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II_∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.     

Serre-Taubes duality for pseudoholomorphic curves

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b₊>1 satisfy the duality Gr(α)=±Gr(κ−α), where κ is Poincaré dual to the canonical class. Extending joint work with Simon Donaldson, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil on X, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods, we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b₊=1 and b₁=0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.     

On the concept of matrix derivative

We discuss how to generalize the concept of vector derivative to matrix derivative, propose two definitions, a ‘broad’ and a ‘narrow’ one, compare the two definitions, and argue in favor of the narrow definition.     

Automorphism groups of positive entropy on minimal projective varieties

We determine the geometric structure of a minimal projective threefold having two ‘independent and commutative’ automorphisms of positive topological entropy, and generalize this result to higher-dimensional smooth minimal pairs (X,G). As a consequence, we give an effective lower bound for the first dynamical degree of these automorphisms of X fitting the ‘boundary case’.     

Quasineutral limit of a time-dependent drift-diffusion-Poisson model for p-n junction semiconductor devices

In this paper the vanishing Debye length limit of the bipolar time-dependent drift-diffusion-Poisson equations modelling insulated semiconductor devices with p-n junctions (i.e., with a fixed bipolar background charge) is studied. For sign-changing and smooth doping profile with ‘good’ boundary conditions the quasineutral limit (zero-Debye-length limit) is performed rigorously by using the multiple scaling asymptotic expansions of a singular perturbation analysis and the carefully performed classical energy methods. The key point in the proof is to introduce a ‘density’ transform and two λ-weighted Liapunov-type functionals first, and then to establish the entropy production integration inequality, which yields the uniform estimate with respect to the scaled Debye length. The basic point of the idea involved here is to control strong nonlinear oscillation by the interaction between the entropy and the entropy dissipation.     

Topological properties of ordinary nematics in 3-space

This paper describes the topologically possible global defect behavior of ordinary nematics in 3-space. It is written for physicists interested in defects of ordered media as well as for topologists, but instead of using an intermediate way of presentation, which might appeal to no one, we first state the result for physicists and then, discussing the proof, turn to mathematicians and physicists who are inclined to read a mathematical paper.     

Invariants of transverse foliations

We construct two invariants for a pair of transverse one-dimensional foliations on the plane. If the set of separatrices is Hausdorff in the space of leaves, the invariant is a distinguished graph. In case there are a finite number of separatrices the invariant is an indexed link.     

Morphisms between complete Riemannian pseudogroups

We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a closure and its maps are C^∞ along the orbit closures. Here, completeness and closure are versions for morphisms of concepts introduced by Haefliger for pseudogroups. This result is applied to approximate foliated maps by smooth ones in the case of transversely complete Riemannian foliations, yielding the foliated homotopy invariance of their spectral sequence. This generalizes the topological invariance of their basic cohomology, shown by El Kacimi-Alaoui-Nicolau. A different proof of the spectral sequence invariance was also given by the second author.     

Symplectic tori in rational elliptic surfaces

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive 2-dimensional homology class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber class of the rational elliptic surface . We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.     

Orientation-reversing actions on lens spaces and Gaussian integers

A finite group action on a lens space L(p,q) has ‘type OR’ if it reverses orientation and has an invariant Heegaard torus whose sides are interchanged by the orientation-reversing elements. In this paper we enumerate the actions of type OR up to equivalence. This leads to a complete classification of geometric finite group actions on amphicheiral lens spaces L(p,q) with p>2. The family of actions of type OR is partially ordered by lifting actions via covering maps. We show that each connected component of this poset may be described in terms of a subset of the lattice of Gaussian integers ordered by divisibility. This results in a correspondence equating equivalence classes of actions of type OR with pairs of Gaussian integers.     

A group based solution strategy for multi-physics simulations in parallel

Multi-physics simulation often requires the solution of a suite of interacting physical phenomena, the nature of which may vary both spatially and in time. For example, in a casting simulation there is thermo-mechanical behaviour in the structural mould, whilst in the cast, as the metal cools and solidifies, the buoyancy induced flow ceases and stresses begin to develop. When using a single code to simulate such problems it is conventional to solve each ‘physics’ component over the whole single mesh, using definitions of material properties or source terms to ensure that a solved variable remains zero in the region in which the associated physical phenomenon is not active. Although this method is secure, in that it enables any and all the ‘active’ physics to be captured across the whole domain, it is computationally inefficient in both scalar and parallel. An alternative, known as the ‘group’ solver approach, involves more formal domain decomposition whereby specific combinations of physics are solved for on prescribed sub-domains. The ‘group’ solution method has been implemented in a three-dimensional finite volume, unstructured mesh multi-physics code, which is parallelised, employing a multi-phase mesh partitioning capability which attempts to optimise the load balance across the target parallel HPC system. The potential benefits of the ‘group’ solution strategy are evaluated on a class of multi-physics problems involving thermo-fluid–structural interaction on both a single and multi-processor systems. In summary, the ‘group’ solver is a third faster on a single processor than the single domain strategy and preserves its scalability on a parallel cluster system.     

A topological minorization for the volume of vector fields on 5-manifolds

A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional.For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.Received: 29 April 2004     

The Hopf invariant of a Haefliger knot

We show that for any given differentiable embedding of the three-sphere in six-space there exists a Seifert surface (in six-space) with arbitrarily prescribed signature. This implies, according to our previous paper, that given such a (6,3)-knot endowed with normal one-field, we can construct a Seifert surface so that the outward normal field along its boundary coincides with the given normal one-field. This aspect enables us to understand the resemblance between Ekholm–Szűcs’ formula for the Smale invariant and a formula in our previous paper for differentiable (6,3)-knots. As a consequence, we show that an immersion of the three-sphere in five-space can be regularly homotoped to the projection of an embedding in six-space if and only if its Smale invariant is even. We also correct a sign error in our previous paper: “A geometric formula for Haefliger knots” [Topology 43: 1425–1447 2004].      

Hopf cyclic cohomology and transverse characteristic classes

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan–Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand–Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of étale foliation groupoids.