A new proof of the odd primary homotopy exponent of spheres

We give a different proof of Cohen, Moore and Neisendorfer’s theorem stating that for odd primes p the homotopy exponent of S ²ⁿ⁺¹ is p ⁿ . This is done using methods recently introduced by Gray and the author to give a new construction of Anick’s fibration.     

Milnor fibration at infinity for mixed polynomials

We study the existence of Milnor fibration on a big enough sphere at infinity for a mixed polynomial f: ?²ⁿ → ?². By using strongly non-degenerate condition, we prove a counterpart of Némethi and Zaharia’s fibration theorem. In particular, we obtain a global version of Oka’s fibration theorem for strongly non-degenerate and convenient mixed polynomials.     

Moser's theorem for lower dimensional tori

Moser's C^?-version of Kolmogorov's theorem on the persistence of maximal quasi-periodic solutions for nearly-integrable Hamiltonian system is extended to the persistence of non-maximal quasi-periodic solutions corresponding to lower-dimensional elliptic tori of any dimension n between one and the number of degrees of freedom. The theorem is proved for Hamiltonian functions of class C^? for any ?>6n+5 and the quasi-periodic solutions are proved to be of class C^p for any p with 2<p<p_* for a suitable p_*=p_*(n,?)>2 (which tends to infinity when ?→∞).     

The metaplectic representation,Weyl operators and spectral theory

Let X be the Grassmannian of Lagrangian subspaces of R²ⁿ and π: Θ → X the bundle of negative half-forms. We construct a canonical imbedding S(Rⁿ)_even → C^∞(Θ) which intertwines the metaplectic representation of M_p(n) on S(Rⁿ) with the induced representation of M_p(n) on C^∞(Θ). This imbedding converts the algebra of Weyl operators into an algebra of pseudodifferential operators and enables us to prove theorems about the spectral properties of Weyl operators by reducing them to standard facts about pseudodifferential operators. For instance we are able to prove a Weyl theorem on the asymptotic growth of eigenvalues with an “optimal” error estimate for such operators and an analogue of the Helton clustering theorem and the Chazarain-Duistermaat-Guillemin trace formula.     

Version Carleman du théorème de l'« edge of the wedge å

J.-P. Rosay has given a C^∞ version of “the edge of the wedge” theorem for non-smooth edges. We give now a “Carleman” version of this theorem, taking on one hand, an edge Rⁿ, and on the other hand, a C³ totally real submanifold of Cⁿ.     

Shape fibrations and strong shape theory

The notion of shape fibration was introduced by Marde?i? and Rushing. In this paper we use ‘fibrant space’ techniques in strong shape theory to prove that every shape fibration p:E → B of compact metric spaces is contained in a map of fibrant spaces p′:E′→B′ which enjoys a certain lifting property and whose homotopy properties reflect the strong shape properties of the map p. Standard methods for studying Hurewicz fibrations are readily applied to the map p' and in this way we obtain a number of strong shape generalizations of results of Marde?i? and Rushing. We also prove the following theorem which answers a question of Rushing: A shape fibration of compact metric spaces which is a strong shape equivalence is an hereditary shape equivalence. Since the converse was known, this gives a characterization of hereditary shape equivalences.     

Exotic analytic structures and Eisenman intrinsic measures

Using Eisenman intrinsic measures we prove a cancellation theorem. This theorem allows to find new examples of exotic analytic structures onC ⁿ under which we understand smooth complex affine algebraic varietiers which are diffeomorphic toR ²ⁿ but not biholomorphic toC ⁿ . We also develop a new method of constructing these structures which enables us to produce exotic analytic structures onC ³ with a given number of hypersurfaces isomorphic toC ² and a family of these structures with a given number of moduli.     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

Analytic discs attached to a generating CR-manifold

Perturbations by analytic discs along generating CR-submanifolds of Cⁿ are considered. In the case all partial indices of a closed pathp in a generating CR-fibration {M(ξ)}_ξ∈∂D are greater or equal to −1 we can completely parametrize all small holomorphic perturbations of the pathp along the fibration {M(ξ)}_ξ∈∂D . In this case we also study the geometry of perturbations by analytic discs and their relation to the conormal bundle of the fibration.     

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set S of simple roots of SL(n,C) is an equivariant fibration X→XS of the complete flag variety X of Cn. To each such fibration we associate an algebra JS of operators on L²(X), or more generally on L²-sections of vector bundles over X. This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X→XS and X→XT, is a compact operator if S∪T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as ‘essential orthogonality of subrepresentations’.     

Another universality property for crumpled cubes

Let C denote a crumpled n-cube in the n-sphere Sⁿ such that every Cantor set in its boundary is tamely embedded in Sⁿ. The main theorem shows C to be universal in the sense that however it is sewn to a crumpled n-cube D of type 2A, a large class containing most of the explicitly described examples, the resultant space is homeomorphic to Sⁿ.     

Best polynomial approximations: A corrected proof

A proof is given of a theorem concerning the best (in the sense of thep-norm) piecewise polynomial approximation of degreen for a function belonging toC ⁿ⁺¹[a,b]. This corrects an earlier, erroneous proof of the same theorem.     

Substantial Riemannian submersions ofS 15 with 7-dimensional fibres

In this paper we show that a substantial Riemannian submersion ofS ¹⁵ with 7-dimensional fibres is congruent to the standard Hopf fibration. As a consequence we prove a slightly weak form of the diameter rigidity theorem for the Cayley plane which is considerably stronger than the very recent radius rigidity theorem of Wilhelm.     

Analogues of Miyachi, cowling-price and Morgan theorems for compact extensions of ℝ n

Let K be a compact subgroup of automorphisms of ? ⁿ . We formulate and prove an analogue of Miyachi’s theorem for the semi-direct product K ? ? ⁿ . This allows us to solve the sharpness problems in the theorem of Cowling-Price and in the L ^p ? L ^q analogue of Morgan theorem for any compact extension of ? ⁿ . These upshots are proved using the representations theory and the Plancherel formula for the group Fourier transform.     

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y² = X³ + a₄X and Y² = X³ + a₆ are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients (_n²ⁿ) (Resp. (_n³ⁿ)) modulo a prime p = 4n + 1 (resp. p = 6n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.     

On representation of Stinespring’s type for n-tuples of completely positive maps in Hilbert C ⋆-modules

Here we prove an analog of the Stinespring’s theorem for n-tuples of completely positive maps in Hilbert C ^?-modules.     

The ordinarity of an isotrivial elliptic fibration

In this paper, we study the ordinarity of an isotrivial elliptic surface defined over a field of positive characteristic. If an isotrivial elliptic fibration ?? : X ?? C is given, X is ordinary when the common fiber of ?? is ordinary and a certain finite cover of the base C is ordinary. By this result, we may obtain the ordinary reduction theorem for some kinds of isotrivial elliptic surfaces defined over a number field.     

Limitation theorems for absolute Nörlund summability

A limitation theorem concerning absolute Nörlund summability was proved recently. In the present paper limitation theorems concerning |N, p, q| summability are proved and corresponding theorems for |E, δ|, |C, α|, |C, α, β|, \(\left| {\bar N,p_n } \right|\) , |N, p _n ^α| methods are deduced as special cases.     

Homology of gauge groups associated with special unitary groups

We study the homology of gauge groups associated with principal SU(n) bundles over the four-sphere. After computing the mod p homology of based gauge groups of SU(n) by combined use of the Serre spectral sequence and the Eilenberg-Moore spectral sequence, we compute the modp homology of gauge groups of SU(n) using the Serre spectral sequence for the evaluation fibration.     

A degeneracy theorem for meromorphic mappings with truncated multiplicities

In this article, we prove a degeneracy theorem for three linearly non-degenerate meromorphic mappings from C n into P N (C), sharing 2N + 2 hyperplanes in general position, counted with multiplicities truncated by 2.