Local rigidity for certain groups of toral automorphisms

Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on ⁿ is locally rigid, i.e., every action of Γ on ⁿ by C^∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C^∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ). Partially supported by NSF grant DMS9011749.     

Rigidity problem for lattices in solvable Lie groups

The paper concerns rigidity problem for lattices in simply connected solvable Lie groups. A lattice Γ⊂G is said to be rigid if for any isomorphism ϕ:Γ→Γ′ with another lattice Γ′⊂G there exists an automorphism which extends ϕ. An effective rigidity criterion is proved which generalizes well-known rigidity theorems due to Malcev and Saito. New examples of rigid and nonrigid lattices are constructed. In particular, we construct: a) rigid lattice Γ⊂G which is not Zariski dense in the adjoint representation ofG, b) Zariski dense lattice Γ⊂G which is not rigid, c) rigid virtually nilpotent lattice Γ in a solvable nonnilpotent Lie groupG.     

Geometric approach to stable homotopy groups of spheres. Kervaire invariants. II

We present an approach to the Kervaire-invariant-one problem. The notion of the geometric (ℤ/2 ⨁ ℤ/2)-control of self-intersection of a skew-framed immersion and the notion of the (ℤ/2 ⨁ ℤ/4)-structure on the self-intersection manifold of a D ₄-framed immersion are introduced. It is shown that a skew-framed immersion ↬ℝ ⁿ , 0 < q ≪ n (in the -range), admits a geometric (ℤ/2 ⨁ ℤ/2)-control if the characteristic class of the skew-framing of this immersion admits a retraction of order q, i.e., there exists a mapping such that this composition → ℝP^∞ is the characteristic class of the skew-framing of f. Using the notion of (ℤ/2 ⨁ ℤ/2)-control, we prove that for a sufficiently large n, n = 2 ^l − 2, an arbitrarily immersed D ₄-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a (ℤ/2 ⨁ ℤ/4)-structure. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 17–41, 2007.     

Toward logarithmic extensions of $$\widehat{s\ell }(2)_k $$ conformal field models

For positive integers p = k + 2, we construct a logarithmic extension of the conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a butterfly resolution of a three-boson realization of . The currents W⁻(z) and W⁺(z) of a W-algebra acting in the kernel are determined by a highest-weight state of dimension 4p − 2 and charge 2p − 1 and by a (θ=1)-twisted highest-weight state of the same dimension 4p − 2 and opposite charge −2p+1. We construct 2p W-algebra representations, evaluate their characters, and show that together with the p−1 integrable representation characters, they generate a modular group representation whose structure is described as a deformation of the (9p−3)-dimensional representation R _p+1⊕ℂ²⊗R _p+1ʕR _p−1⊕ℂ² R _p−1⊕ℂ³ R _p−1, where R _p−1 is the SL(2, ℤ)-representation on integrable-representation characters and R _p+1 is a (p+1)-dimensional SL(2, ℤ)-representation known from the logarithmic (p, 1) model. The dimension 9p − 3 is conjecturally the dimension of the space of torus amplitudes, and the ℂⁿ with n = 2 and 3 suggest the Jordan cell sizes in indecomposable W-algebra modules. We show that under Hamiltonian reduction, the W-algebra currents map into the currents of the triplet W-algebra of the logarithmic (p, 1) model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 291–346, December, 2007.     

Finite-dimensional crystals B 2,s for quantum affine algebras of type D (1) n

The Kirillov–Reshetikhin modules W^r,s are finite-dimensional representations of quantum affine algebras U’_q labeled by a Dynkin node r of the affine Kac–Moody algebra and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^2,s corresponding to W^2,s for the algebra of type D⁽¹⁾_n. 2000 Mathematics Subject Classification Primary—17B37; Secondary—81R10 Supported in part by the NSF grants DMS-0135345 and DMS-0200774.     

The Littlewood-Gowers problem

The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions ƒ ∈ L ²(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space endowed with the norm , and generalize this to the approximate Fourier transform on Bohr sets. As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/3−ɛ; we improve this to ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ _A ‖ _A(ℤ/pℤ) ≪ log p.     

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.     

Automorphic L-Functions in the Weight Aspect

Let S_k(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let S_k(Γ)⁺ be the set of all Hecke eigenforms from this space with the first Fourier coefficient a_f(1) = 1. For f ∈ S_k(Γ)⁺, consider the Hecke L-function L(s, f). Let

Up–down representations and ergodic theory in nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G.     

Maximal regularity and asymptotic behavior of solutions for the Cahn–Hilliard equation with dynamic boundary conditions

This paper deals with the Cahn–Hilliard equation subject to the boundary conditions and the initial condition ψ(0,x) = ψ₀(x) where J = (0,∞), and Ω ⊂ ℝ ⁿ is a bounded domain with smooth boundary Γ = ∂ G, n≤ 3, and Γ _s ,σ _s ,g _s > 0, h are constants. This problem has already been considered in the recent paper of R. Racke and S. Zheng (The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Diff. Eq. 8, 83–110, 2003), where global existence and uniqueness were obtained. In this paper we first obtain results on the maximal L _p -regularity of the solution. We then study the asymptotic behavior of the solution of this problem and prove the existence of a global attractor. Mathematics Subject Classification (2000) 82C26, 35B40, 35B65, 35Q99     

Extremals for Logarithmic Hardy-Littlewood-Sobelov Inequalities on Compact Manifolds

Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g Γ, denote the area element by dV and the Laplace–Beltrami operator by Δ _g . We define the Robin mass m(x) at the point x M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of Δ _g ⁻¹ is then defined by trace Δ⁻¹ = ∫ _M m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let be the Laplace–Beltrami operator on the round sphere of volume V. We show that if there exists g Γ with trace Δ _g ⁻¹ < trace then the minimum of trace Δ⁻¹ over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace Δ⁻¹ over Γ is equal to trace . In fact we prove these results in the general setting where M is an n-dimensional closed, connected manifold and the Laplace–Beltrami operator is replaced by any non-negative elliptic operator A of degree n which is conformally covariant in the sense that for the metric g we have . In this case the role of is assumed by the Paneitz or GJMS operator on the round n-sphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the Onofri–Beckner theorem. Received: February 2006, Accepted: March 2006     

Riesz multiwavelet bases generated by vector refinement equation

In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L ₂(ℝ ^s ). Suppose ψ = (ψ¹,..., ψ ^r ) ^T and are two compactly supported vectors of functions in the Sobolev space (H ^μ(ℝ ^s )) ^r for some μ > 0. We provide a characterization for the sequences {ψ _jk ^l : l = 1,...,r, j ε ℤ, k ε ℤ ^s } and to form two Riesz sequences for L ₂(ℝ ^s ), where ψ _jk ^l = m ^j/2ψ ^l (M ^j ·−k) and , M is an s × s integer matrix such that lim _n→∞ M ⁻ⁿ = 0 and m = |detM|. Furthermore, let ϕ = (ϕ¹,...,ϕ ^r ) ^T and be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, and M, where a and are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψ^ν = (ψ^ν1,...,ψ^νr ) ^T and , ν = 1,..., m − 1 such that two sequences {ψ _jk ^νl : ν = 1,..., m − 1, l = 1,...,r, j ε ℤ, k ε ℤ ^s } and { : ν=1,...,m−1,ℓ=1,...,r, j ∈ ℤ, k ∈ ℤ ^s } form two Riesz multiwavelet bases for ^L ₂(ℝ ^s ). The bracket product [f, g] of two vectors of functions f, g in (L ₂(ℝ ^s )) ^r is an indispensable tool for our characterization. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123)     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

Root differences for A n

Suppose W is a Weyl group with Φ = Φ(W) a root system of W. The set D of root differences is given by D = {α − β : α, β, ∈ Φ}. We define t(Φ) to be the maximum exponent of the torsion subgroup of for any In this article we show that if W is of type A_n, then t(Φ) = 2ⁿ. Received: 25 November 2004     

A geometrical problem arising in a signal restoration algorithm

Let ℤ_2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ_2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ_2N→ℝ is given by φ_f(k)=arg for k ∈ ℤ_2N, where denotes the discrete Fourier transforms of f. For a fixed s∈L let K_s denote the cone of all f:ℤ_2N→ℝ which satisfy φ_f ≡ φ_s and let M_s be its linear span. The angle α_s between M_s and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove the following conjectures of Urieli et al. [7] who verified them for the N≤3 case:

Estimates for the number of rational points on convex curves and surfaces

Let Γ ⊂ ℝ^d be a bounded strictly convex surface. We prove that the number k_n(Γ) of points of Γ that lie on the lattice satisfies the following estimates: lim inf k_n(Γ)/n^d−2 < ∞ for d ≥ 3 and lim inf k_n(Γ)/log n < ∞ for d = 2. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 174–189.     

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

We consider the Riemann–Hilbert problem in the following setting: find a function whose boundary values ϕ⁺(t) satisfy the condition a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, and K ^{p( · )}(D;ω) is the set of functions ϕ(z) representable in the form , where ω(z) is a weight function and (K _Γφ )(z) is a Cauchy type integral whose density φ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewise-Lyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-H?lder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points. Submitted: May 13, 2007. Revised: August 8, 2007 and August 28, 2007. Accepted: November 8, 2007.     

The bieri-strebel invariant and homological finiteness conditions for metabelian groups

A group Γ has type F P_n if a trivial ℤΓ-module ℤ has a projective resolution P:…P_n → … → P₁ → P₀ → ℤ in which ℤΓ-module P_n,…P₁, P₀ are finitely generated. Let the finitely generated group Γ be a split extension of the Abelian group M by an Abelian group Q, suppose M is torsion free, and assume Γ∈F P_m, m≥2. Then the invariant ∑ _c ^M is m-tame. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 194–218, March–April, 1997.