Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation on a closed manifold M, it is known that is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form (relatively to a suitable riemannian metric μ) is zero (cf. álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group , where (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471, 1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF). J. I. Royo Prieto was partially supported by EHU06/05, by a PostGrant from the Basque Government and by the MCyT of the Spanish Government. R. Wolak was partially supported by the KBN grant 2PO3A 021 25.     

Finite maximal orientation reversing group actions on handlebodies and 3-manifolds

We study arbitrary (that is not necessarily orientation preserving) finite group actions on 3-dimensional orientable or nonorientable handlebodies of genus g. For g>1, the maximal possible order is 24(g−1); we characterize the corresponding groups of this order and also the occuring quotient orbifolds. Then we use this to study finite group actions of large order (with respect to the equivariant Heegaard genus g) on closed 3-manifolds, again concentrating on the maximal case of order 24(g−1). Our results extend corresponding results in the orientation preserving setting. Whereas for arbitrary finite group actions on handlebodies much more types of quotient orbifolds occur than in the orientation preserving case, it turns out that for closed 3-manifolds the situation is quite rigid, in contrast to the orientation preserving case where one has many possibilities to construct manifolds with large group actions.     

Rational surfaces in algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. We study modulo 2 homology classes represented by rational algebraic surfaces in X, as X runs through the class of all algebraic models of M. Received: 16 June 2007     

Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds

We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.     

Homogeneous symmetric means of two variables

Summary.  Let be given continuous functions on the interval I such that g ≠ 0, and is strictly monotonic (thus invertible) on I. Taking an increasing nonconstant function μ on [0, 1]

Uniform algebras on the sphere invariant under group actions

It is shown under certain conditions that a uniform algebra on the unit sphere S in C ² that is invariant under the action of the 2-torus must be C(S). Contrasting with this, an example is presented showing that the statement becomes false when 2 is replaced by n > 2. It is also shown that C(M) is the only uniform algebra on a smooth manifold M that is invariant under a transitive Lie group action on its maximal ideal space. The results presented answer a question raised by Ronald Douglas in connection with a conjecture of William Arveson.     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

On the quantum homology algebra of toric Fano manifolds

We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.      

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

Scattering of an incident acoustic wave by an elastic sphere in a shallow water

The scattering of acoustic waves by an elastic sphere in a shallow ocean wave guide is investigated taking into account the shear waves which can exist in addition to compressional waves in scatterers of solid material. Expressions for the scattered waves are given. Numerical values for a quantity called the farfield form function for various depth are presented in graphical forms.      

Invariants of knot diagrams

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams. J. Hass was partially supported by an NSF grant.     

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.     

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

From Lucid to TransLucid: Iteration, Dataflow, Intensional and Cartesian Programming

We present the development of the Lucid language from the Original Lucid of the mid-1970s to the TransLucid of today. Each successive version of the language has been a generalisation of previous languages, but with a further understanding of the problems at hand. The Original Lucid (1976), originally designed for purposes of formal verification, was used to formalise the iteration in while-loop programs. The pLucid language (1982) was used to describe dataflow networks. Indexical Lucid (1987) was introduced for intensional programming, in which the semantics of a variable was understood as a function from a universe of possible worlds to ordinary values. With TransLucid, and the use of contexts as firstclass values, programming can be understood in a Cartesian framework.      

Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation

Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation are given. Applications for the trapezoid and mid-point inequalities are also provided. Received: 19 May 2008     

Free membrane eigenvalues

For a simply connected and normalized domain D in the plane it was proven by Pólya and Schiffer in 1954 for the fixed membrane eigenvalues

Riemannian manifolds carrying a pair of skew symmetric conformal vector fields

We deal with a Riemannian manifoldM carrying a pair of skew symmetric conformal vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case,M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved. Supported by a JSPS postdoctoral fellowship.     

Critical Nonlinear Nonlocal Equations on a Half-Line

We study nonlinear nonlocal equations on a half-line in the critical case