Gently killing S-spaces

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^ℵ ₀<2^ℵ ₁. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size ℵ₁, answering a question of the second author [9]. The first author would like to thank The Hebrew University of Jerusalem for their support while the research in this paper was being carried out. The research of the second author was partially supported by NSF Grant DMS-9322613. The research of the third author was partially supported by NSF grant DMS-9704477 and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. This is publication number 690 in the list of the third author.     

The X-Ray Transform for a Generic Family of Curves and Weights

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T ^* M. Second author was partly supported by NSF Grant DMS-0400869; third author was partly supported by NSF and a Walker Family Endowed Professorship.     

A General Fredholm Theory II: Implicit Function Theorems

This is the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. In general, these spaces are not locally homeomorphic to open sets in Banach spaces. The current paper develops the Fredholm theory in M-polyfold bundles. It consists of a transversality and a perturbation theory. In upcoming papers the generalized Fredholm theory will be applied to the Floer Theory, the Gromov–Witten Theory and the Symplectic Field Theory H.H.’s research partially supported by NSF grant DMS-0603957. K.W.’s research partially supported by NSF grant DMS-0606588.     

Harmonic Functions,Entropy, and a Characterization of the Hyperbolic Space

Let (M ⁿ ,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ ₀ of its universal covering satisfies λ ₀≤(n−1)²/4. We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. The author was partially supported by NSF Grant 0505645.     

The Gauss-Bonnet theorem for vector bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles. Dedicated to the memory of Philip Bell Research partially supported by NSF grant DMS-9703852.     

Wiggly sets and limit sets

We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that ifG is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1. The first author is partially supported by NSF Grant DMS 95-00577 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF grant DMS-94-23746.     

An operator onL pwithout best compact approximation

We construct, for 1<p<∞,p ≠ 2, an operator onL ^pwhose distance to the space of compact operators onL ^pis not attained. We also show that the identity operator onL ^p,p ≠ 1,2, ∞ has a unique best compact approximation. Research partially supported by NSF grant DMS-8201635.     

Isometry of Riemannian manifolds to spheres

Summary Some known conditions for a compact Riemannian n-manifold Mⁿ, n>2, which has constant scalar curvature R and admits an infinitesimal nonisometric conformal transformation, to be isometric to an n-sphere are generalized to manifolds with nonconstant R. Entrata in Redazione il 20 giugno 1972. The work of the second author was partially supported by NSF grant GP-11965.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

Isometric actions of SLn(ℝ) ⋉ ℝn on Lorentz manifolds

We prove that a locally faithful, isometric action of SL_n(ℝ) ⋉ ℝⁿ on a connected Lorentz manifold must be a proper action. This provides an essential step toward classifying nonproper isometry groups of noncompact Lorentz manifolds. The first author was supported in part by NSF grant DMS-9703480.     

Compact deformations of Fuchsian groups

A conformal map Φ on the unit disk is called a compact deformation of a Fuchsian groupG if Φ has a quasiconformal extension to the planeh which conjugatesG to a Kleinian group G′ and the dilatation ofh is compactly supported moduloG. We show that for such deformations δ = dim(∧(G′)) = dim(∧_c(G′)) (if δ ≥1) and the image of ∧_e = ∧ ∧_c is contained in a countable union of rectifiable curves and has zero length iffG is divergence type. The first author is partially supported by NSF Grant DMS 01-03626. The second author is partially supported by NSF grant DMS-99-71311.     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

On deformations of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in compact Lie groups

We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular, we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(F _n ) on such variety when n ≥ 3. The author was partially supported by NSF grant DMS-0404557, BSF grant 2004010, and the ‘Finite Structures’ Marie Curie Host Fellowship, carried out at the Alfréd Rényi Institute of Mathematics in Budapest.     

Large deviations and stochastic flows of diffeomorphisms

Summary Previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold M are extended and then applied to the analysis of the Lyapunov exponents of a stochastic flow of diffeomorphisms of M. An approximation argument relates these results to the behavior near the diagonal Δ in M ² of the associated two point motion. Finally it is shown, under appropriate non-degeneracy conditions, that the two-point motion is ergodic on M ²-Δ if the top Lyapunov exponent is positive. At the period when this research was initiated, both authors where guests of the I.M.A. in Minneapolis. The first author was at Aberdeen University, Scotland when this article was prepared. Throughout the period of this research, the second author has been partially supported by N.S.F. grant DMS-8611487 and ARO grant DAAL03-86-K-171     

Quasiequational Theories of Flat Algebras

We prove that finite flat digraph algebras and, more generally, finite compatible flat algebras satisfying a certain condition are finitely q-based (possess a finite basis for their quasiequations). We also exhibit an example of a twelve-element compatible flat algebra that is not finitely q-based. The first author was partially supported by the grant # 201/02/0594 of the Grant Agency of the Czech Republic, and by the Institutional grant MSM0021620839; the second author was partially supported by the grant No. Tn37877 of the Hungarian National Foundation for Scientific Research (OTHA); the third author was supported by the NSF grant # DMS-9971352.     

Central Value Of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindel?f hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting. The first author was partially supported by NSF grant DMS-0070762. The second author was partially supported by NSF grant DMS-0355285. Received: July 2005 Accepted: August 2005     

Coloring complexes and arrangements

Steingrimsson’s coloring complex and Jonsson’s unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ℬ _n arrangements. The first author was supported by NSF grant DMS-0500638. The second author was supported by NSF grant DMS-0245623.     

On minimal,strongly proximal actions of locally compact groups

Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie groupG on homogeneous spacesG/Q, whereQ ⊂G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note we show that ifX is a compact manifold with a faithful boundary action of some locally compact groupH, then (under some mild regularity assumption) the groupH, the spaceX, and the action split into a direct product of a semi-simple Lie groupG acting onG/Q and a boundary action of a discrete countable group. The author was partially supported by NSF grants DMS-0049069, 0094245 and GIF grant G-454-213.06/95.     

Bessel identities in the waldspurger correspondence over the real numbers

We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL ₂(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL ₂(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. Research of first author was partially supported by NSF grant DMS-0070762. Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003.     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077