Normal families of meromorphic functions and shared values

Let k be a positive integer, b?≠ 0 be a finite complex number, let P be a polynomial with either deg P ≥ 3 or deg P = 2 and P having only one distinct zero, and let ${\mathcal{F}}$ be a family of functions meromorphic in a domain D, all of whose zeros have multiplicities at least k. If, each pair of functions f and g in ${\mathcal{F}, P(f)f^{(k)}}$ and P(g)g ^(k) share b in D, then ${\mathcal{F}}$ is normal in D.     

Normality of meromorphic functions whose derivatives have 1-points

Let k be a positive integer and let ${\mathcal F}Let k be a positive integer and let F{\mathcal F} be a family of functions meromorphic in a plane domain D, all of whose zeros have multiplicity at least k + 3. If there exists a subset E of D which has no accumulation points in D such that for each function f ? F{f\in\mathcal F}, f ^(k)(z) − 1 has no zeros in D\E{D\setminus E}, then F{\mathcal F} is normal. The number k + 3 is sharp. The proof uses complex dynamics.     

A normality criterion involving rotations and dilations in the argument

We show that a family F\mathcal{F} of analytic functions in the unit disk \mathbbD{\mathbb{D}} all of whose zeros have multiplicity at least k and which satisfy a condition of the form

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}

Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP²{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.     

Sharing Set and Normal Families of Entire Functions

Let ${\mathcal{F}}$ be a family of holomorphic functions defined in a domain ${\mathcal{D}}$ , let k( ≥ 2) be a positive integer, and let S = {a, b}, where a and b are two distinct finite complex numbers. If for each ${f \in \mathcal{F}}$ , all zeros of f(z) are of multiplicity at least k, and f and f ^(k) share the set S in ${\mathcal{D}}$ , then ${\mathcal{F}}$ is normal in ${\mathcal{D}}$ . As an application, we prove a uniqueness theorem.     

Finding Convex Sets in Convex Position

Let F{\cal{F}} denote a family of pairwise disjoint convex sets in the plane. F{\cal{F}} is said to be in convex position, if none of its members is contained in the convex hull of the union of the others. For any fixed k 3 5k\ge5, we give a linear upper bound on P_k(n)P_k(n), the maximum size of a family F{\cal{F}} with the property that any k members of F{\cal{F}} are in convex position, but no n are.     

Topology and Smooth Structure for Pseudoframes

Given a closed subspace ${\mathcal{S}}Given a closed subspace S{\mathcal{S}} of a Hilbert space H{\mathcal{H}}, we study the sets F_S{\mathcal{F}_\mathcal{S}} of pseudo-frames, CF_S{\mathcal{C}\mathcal{F}_\mathcal{S}} of commutative pseudo-frames and \mathfrakX_S{\tiny{\mathfrak{X}}_{\mathcal{S}}} of dual frames for S{\mathcal{S}}, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({f_n}_{n ? \mathbbN},{h_n}_{n ? \mathbbN}){(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})},

F:l2? H,     F({cn}n ? \mathbbN )=?n cn fn,F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,      10. Triangulating Smooth Submanifolds with Light Scaffolding      Jean-Daniel Boissonnat  Arijit Ghosh 《Mathematics in Computer Science》2010,4(4):431-461 We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in \mathbbRd{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of \mathbbR 3{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of \mathbbR d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e2){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k).      11. Deformations of circle-valued Morse functions on surfaces      S. I. Maksymenko 《Ukrainian Mathematical Journal》2011,62(10):1577-1584 Let M be a smooth connected orientable compact surface and let Fcov ( M,S1 ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) be a space of all Morse functions f : M → S 1 without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S 1 is either a constant map or a covering map. The space Fcov ( M,S1 ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) is endowed with the C ∞-topology. We present the classification of connected components of the space Fcov ( M,S1 ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) . This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M.      12. Singularity Spectrum of Generic ��-H?lder Regular Functions After Time Subordination      Zolt��n Buczolich  St��phane Seuret 《Journal of Fourier Analysis and Applications》2011,17(3):457-485 A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by B1a{\mathcal {B}}_{1}^{\alpha} the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function g ? B1ag\in {\mathcal {B}}_{1}^{\alpha} one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space Ea=M×B1a\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha} equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ α . In particular we determine the generic H?lder singularity spectrum of g○f.      13. Normal families and shared functions of meromorphic functions      Jun-Fan Chen  Ming-Liang Fang 《Israel Journal of Mathematics》2010,180(1):129-142 Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f (k) share 0, and |f(z)| ≥ M whenever f (k)(z) = h(z), then F is normal in D. The condition that f and f (k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f (k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f (k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.      14. On thin-complete ideals of subsets of groups      T. Banakh  N. Lyaskovska 《Ukrainian Mathematical Journal》2011,63(6):865-879 Let F ì PG \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of PG {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g n ) n∈ω of nonzero elements of G, there is n ∈ ω such that ?i0, ?, in ? { 0,  1 } g0i0 ?gninA ? F . \bigcap\limits_{{i_0}, \ldots, {i_n} \in \left\{ {0,\;1} \right\}} {g_0^{{i_0}} \ldots g_n^{{i_n}}A \in \mathcal{F}} .      15. A Splitting of the Frobenius Morphism on the Whole Algebra of Distributions of SL 2      Michel Gros 《Algebras and Representation Theory》2012,15(1):109-118 We define, over k = \BbbFpk = {\Bbb{F}}_{p}, a splitting of the Frobenius morphism Fr : \textDist (G) ? \textDist (G)Fr : {\text{Dist}}\,(G) \rightarrow {\text{Dist}}\,(G) on the whole \textDist (G){\text{Dist}}\,(G), the algebra of distributions of the k-algebraic group G: = SL 2. This splitting is compatible (and lifts) the theory of Frobenius descent for arithmetic D{\cal{D}}-modules over X:=\BbbPk1X:={\Bbb{P}}_{k}^{1}.      16. Desingularization of singular Riemannian foliation      Marcos M. Alexandrino 《Geometriae Dedicata》2010,149(1):397-416 Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]n/[^(F)]n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.      17. Modular Lie algebras and the Gelfand–Kirillov conjecture      Alexander Premet 《Inventiones Mathematicae》2010,181(2):395-420 Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg\Bbbk=\mathfrakg\mathbbZ?\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbFp{\mathbb{F}}_{p}. Let G\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G\Bbbk)=\mathfrakg\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg\Bbbk)G\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B n , n≥3, D n , n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B n , n≥3, D n , n≥4, E6, E7, E8 or F4, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.      18. The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic      Terence Tao  Tamar Ziegler 《Annals of Combinatorics》2012,16(1):121-188 We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function f : V ? \mathbbC{f : V \rightarrow \mathbb{C}} on a finite-dimensional vector space V over a finite field \mathbbF{\mathbb{F}} has large Gowers uniformity norm ||f||Us+1(V){{\parallel{f}\parallel_{U^{s+1}(V)}}} , then there exists a (non-classical) polynomial P: V ? \mathbbT{P: V \rightarrow \mathbb{T}} of degree at most s such that f correlates with the phase e(P) = e 2πiP . This conjecture had already been established in the “high characteristic case”, when the characteristic of \mathbbF{\mathbb{F}} is at least as large as s. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson [3], together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author [22] and of Kaufman and Lovett [28].      19. {\mathcal{F}}_\phi and {\mathcal{F}}_g Classes      Lino F. Reséndis O.  Luis M. Tovar S. 《Advances in Applied Clifford Algebras》2010,20(1):85-110 In this paper we give a generalization of the Zhao F(p, q, s)-spaces by using operators instead of functions. In this way we unify and simplify several important results about the classic spaces Dp, Qp{\mathcal{Q}}_{p} ,Bα, etc.      20. Augmented Teichmüller spaces and orbifolds      Vladimir Hinich  Arkady Vaintrob 《Selecta Mathematica, New Series》2010,16(3):533-629 We study complex analytic properties of the augmented Teichmüller spaces [`(T)]g,n{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces Tg,n{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike Tg,n{\mathcal{T}_{g,n}}, the space [`(T)]g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Triangulating Smooth Submanifolds with Light Scaffolding

We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in \mathbbR^d{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of \mathbbR ³{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of \mathbbR ^d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e^-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e²){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e^-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k).     

Deformations of circle-valued Morse functions on surfaces

Let M be a smooth connected orientable compact surface and let F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) be a space of all Morse functions f : M → S ¹ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S ¹ is either a constant map or a covering map. The space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) is endowed with the C ^∞-topology. We present the classification of connected components of the space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) . This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M.     

Singularity Spectrum of Generic ��-H?lder Regular Functions After Time Subordination

A question of Yves Meyer motivated the research concerning “time” subordinations of real functions. Denote by B₁^a{\mathcal {B}}_{1}^{\alpha} the metric space of functions with Lipschitz constant 1 defined on [0,1], equipped with the complete metric defined via the supremum norm. Given a function g ? B₁^ag\in {\mathcal {B}}_{1}^{\alpha} one obtains a time subordination of g simply by considering the composite function Z=g○f, where f∈ℳ:={f:f(0)=0, f(1)=1 and f is a continuous nondecreasing function on [0,1]}. The metric space E^a=M×B₁^a\mathcal {E}^{\alpha}=\mathcal {M}\times {\mathcal {B}}_{1}^{\alpha} equipped with the product supremum metric is a complete metric space. In this paper for all α∈[0,1) multifractal properties of g○f are investigated for a generic (typical) element (f,g)∈ℰ ^α . In particular we determine the generic H?lder singularity spectrum of g○f.     

Normal families and shared functions of meromorphic functions

Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f ^(k) share 0, and |f(z)| ≥ M whenever f ^(k)(z) = h(z), then F is normal in D. The condition that f and f ^(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f ^(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f ^(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.     

On thin-complete ideals of subsets of groups

Let F ì P_G \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t^*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of P_G {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g _n ) _n∈ω of nonzero elements of G, there is n ∈ ω such that

A Splitting of the Frobenius Morphism on the Whole Algebra of Distributions of SL 2

We define, over k = \BbbF_pk = {\Bbb{F}}_{p}, a splitting of the Frobenius morphism Fr : \textDist (G) ? \textDist (G)Fr : {\text{Dist}}\,(G) \rightarrow {\text{Dist}}\,(G) on the whole \textDist (G){\text{Dist}}\,(G), the algebra of distributions of the k-algebraic group G: = SL ₂. This splitting is compatible (and lifts) the theory of Frobenius descent for arithmetic D{\cal{D}}-modules over X:=\BbbP_k¹X:={\Bbb{P}}_{k}^{1}.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg_\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg_\Bbbk=\mathfrakg_\mathbbZ?_\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbF_p{\mathbb{F}}_{p}. Let G_\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G_\Bbbk)=\mathfrakg_\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg_\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg_\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg_\Bbbk)^G_\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)^\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.     

The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic

We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function f : V ? \mathbbC{f : V \rightarrow \mathbb{C}} on a finite-dimensional vector space V over a finite field \mathbbF{\mathbb{F}} has large Gowers uniformity norm ||f||_U^s+1(V){{\parallel{f}\parallel_{U^{s+1}(V)}}} , then there exists a (non-classical) polynomial P: V ? \mathbbT{P: V \rightarrow \mathbb{T}} of degree at most s such that f correlates with the phase e(P) = e ^2πiP . This conjecture had already been established in the “high characteristic case”, when the characteristic of \mathbbF{\mathbb{F}} is at least as large as s. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson [3], together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author [22] and of Kaufman and Lovett [28].     

{\mathcal{F}}_\phi and {\mathcal{F}}_g Classes

In this paper we give a generalization of the Zhao F(p, q, s)-spaces by using operators instead of functions. In this way we unify and simplify several important results about the classic spaces D_p, Q_p{\mathcal{Q}}_{p} ,B^α, etc.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.