Central extensions of gerbes

We introduce the notion of central extension of gerbes on a topological space X. We show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also discuss pronilpotent gerbes. These results are used in the paper [15] by A. Yekutieli to study twisted deformation quantization on algebraic varieties.     

Compact embeddings and proper mappings

Sufficient conditions are given to assert that a C¹-proper mapping defined on a compact embedding between Banach spaces over K has a fixed point. The proof of the result is based upon continuation methods.     

Jumps in cohomology and free group actions

A discrete group G has periodic cohomology over R if there is an element in a cohomology group cup product with which it induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed that if , then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for -groups and solvable groups. We also find necessary conditions for free and proper group actions on finite dimensional complexes homotopy equivalent to closed, orientable manifolds.     

Extensions for finite Chevalley groups I

Let G be a connected semisimple algebraic group defined and split over the field with p elements, and k be the algebraic closure of . Assume further that G is almost simple and simply connected and let be the finite Chevalley group consisting of -rational points of G where q=p^r for a non-negative integer r. In this paper, formulas are found relating extensions between simple -modules and extensions over G (considered as an algebraic group over k). One of these formulas, which only holds for primes p?3(h−1) (where h is the Coxeter number of G), is then used to show the vanishing of self-extensions between simple -modules except for certain simple modules when r=1 and the underlying root system is of type A₁ or C_n.     

On deformations of Hamiltonian actions

In this paper we generalize to coisotropic actions of compact Lie groups a theorem of Guillemin on deformations of Hamiltonian structures on compact symplectic manifolds. We show how one can reconstruct from the moment polytope the symplectic form on the manifold. Received: 21 March 2006     

Dégénérescence de séries d’Eisenstein hyperboliques

In this work, we deal with hyperbolic Eisenstein series and more particularly, given a degenerating family S _l (l ≥ 0) of Riemann Surfaces with their canonical hyperbolic metrics, we work out the degeneration of hyperbolic Eisenstein series associated to the pinching geodesics in S _l . Our principal results are theorems 2.2, 4.1 et 4.2.     

Infinitely many leaf-wise intersections on cotangent bundles

If the homology of the free loop space of a closed manifold B$B$ is infinite dimensional then generically there exist infinitely many leaf-wise intersection points for fiberwise star-shaped hypersurfaces in T^∗B$T^{\ast }B$. We illustrate this in the case of the restricted three body problem.     

Good coverings for section spaces of fibre bundles

In this note, existence of good coverings for section spaces of smooth fibre bundles is shown by resorting to the Riemannian geometry of the L²-metric, via the construction of strongly convex neighbourhoods, in close analogy with the finite dimensional case.     

The conformal Killing equation on forms—prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on (k−?)-forms for various integers ?. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.     

Nonlinear flows and rigidity results on compact manifolds

This paper is devoted to rigidity results for some elliptic PDEs and to optimal constants in related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. The largest value of this parameter provides an estimate for the optimal constant in the corresponding interpolation inequality. Our approach relies on a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works on rigidity. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constants in the interpolation inequalities.     

Linear instability of relative equilibria for n-body problems in the plane

Following Smale, we study simple symmetric mechanical systems of n point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion.     

Fredholm operators,evolution semigroups,and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ  -periodic evolutionary process _{{U(t,s)}t≥s}${\{U(t,s)\}}_{t\ge s}$ on the space _Pτ(X)$P_{\tau }(X)$ of all τ-periodic continuous X  -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?)$Lp=-?F(p,?)$ under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0)$U(\tau ,0)$. The proof is based on a new decomposition of the space _Pτ(X)$P_{\tau }(X)$ by constructing a right inverse of L.     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Nonpositive curvature: A geometrical approach to Hilbert-Schmidt operators

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that GM, the Banach-Lie group generated by M, acts isometrically and transitively on M. Moreover, GM admits a polar decomposition relative to M, namely GM?M×K as Hilbert manifolds (here K is the isotropy of p=1 for the action ), and also GM/K?M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism NM?Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed ea, we obtain ea=exevex with ex∈M and v orthogonal to M at p=1. As a corollary we obtain decompositions for the full group of invertible elements G?M×exp(T₁M^⊥)×K.     

Bounds for ratios of the membrane eigenvalues

For a bounded planar region in R², we obtain the ratios of lower order eigenvalues of Laplace operator. Combining our results with the recursive formula in Cheng and Yang (2007) [11], we can obtain better upper bound of the (k+1)-th (k?3) membrane eigenvalues.     

Coronae of product spaces and the coarse Baum–Connes conjecture

We study the coarse Baum–Connes conjecture for product spaces and product groups. We show that a product of CAT(0) groups, polycyclic groups and relatively hyperbolic groups which satisfy some assumptions on peripheral subgroups, satisfies the coarse Baum–Connes conjecture. For this purpose, we construct and analyze an appropriate compactification and its boundary, “corona”, of a product of proper metric spaces.     

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A$A$ and B$B$ are nontrivial conjugacy classes, then AB$AB$ is not a conjugacy class. We also prove that if G$G$ is a finite simple group of Lie type and A$A$ and B$B$ are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB$AB$ is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for p$p$-elements with p≥5$p\ge 5$.     

Nonlinear biharmonic equations with negative exponents

In this paper, we study global positive C⁴ solutions of the geometrically interesting equation: Δ²u+u−q=0 with q>0 in R³. We will establish several existence and non-existence theorems, including the classification result for q=7 with exactly linear growth condition.     

Multiplicity results for problems involving the Hardy-Sobolev operator via Morse theory

We establish some multiplicity results for a class of boundary value problems involving the Hardy-Sobolev operator using Morse theory.