Proper Holomorphic Embeddings of Riemann Surfaces with Arbitrary Topology into ℂ2

We prove that given an open Riemann surface $\mathcal{N}$ of arbitrary (finite or infinite) topology, there exists an open domain $\mathcal{M}\subset \mathcal{N}$ homeomorphic to $\mathcal{N}$ which properly holomorphically embeds in ?². Furthermore, $\mathcal{M}$ can be chosen with hyperbolic conformal type. In particular, any open orientable surface M admits a complex structure $\mathcal{C}$ such that $(M,\mathcal{C})$ can be properly holomorphically embedded into ?².     

The polynomial multidimensional Szemerédi Theorem along shifted primes

If $\vec q_1 ,...,\vec q_m $ : ? → ? ^? are polynomials with zero constant terms and E ? ? ^? has positive upper Banach density, then we show that the set E ∩ (E ? $\vec q_1 $ (p ? 1)) ∩ … ∩ (E ? $\vec q_m $ (p ? 1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.     

A Triangulation of ?P 3 as Symmetric Cube of S 2

The symmetric group $\operatorname{Sym}(d)$ acts on the Cartesian product (S ²) ^d by coordinate permutation, and the quotient space $(S^{2})^{d}/\operatorname{Sym}(d)$ is homeomorphic to the complex projective space ?P ^d . We used the case d=2 of this fact to construct a 10-vertex triangulation of ?P ² earlier. In this paper, we have constructed a 124-vertex simplicial subdivision $(S^{2})^{3}_{124}$ of the 64-vertex standard cellulation $(S^{2}_{4})^{3}$ of (S ²)³, such that the $\operatorname{Sym}(3)$ -action on this cellulation naturally extends to an action on $(S^{2})^{3}_{124}$ . Further, the $\operatorname{Sym}(3)$ -action on $(S^{2})^{3}_{124}$ is ??good??, so that the quotient simplicial complex $(S^{2})^{3}_{124}/\operatorname{Sym}(3)$ is a 30-vertex triangulation $\mathbb{C}P^{3}_{30}$ of ?P ³. In other words, we have constructed a simplicial realization $(S^{2})^{3}_{124} \to\mathbb{C} P^{3}_{30}$ of the branched covering (S ²)³???P ³.     

Fixed points of automorphisms preserving the length of words in free solvable groups

Let ?? be an automorphism of prime order p of the free group F _n . Suppose ?? has no fixed points and preserves the length of words. By ?? :=??? ^(m) we denote the automorphism of the free solvable group ${F_{n}/F_n^{(m)} }$ induced by ??. We show that every fixed point of ?? has the form ${cc^{\sigma} \ldots c^{\sigma^{p-1}}}$ , where ${c\in F_n^{(m-1)}/F_n^{(m)}}$ . This is a generalization of some known results, including the Macedo??ska?CSolitar Theorem [10].     

On Surface Attractors and Repellers in 3-Manifolds

We show that if f: M ³ → M ³ is an A diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ with support $M_\mathcal{B}^2$ , then $\mathcal{B} = M_\mathcal{B}^2$ and there exists a k ≥ 1 such that (1) $M_\mathcal{B}^2$ is the disjoint union M ₁ ² ? ? ? M _k ² of tame surfaces such that each surface M _i ² is homeomorphic to the 2-torus T ²; (2) the restriction of f ^k to M _i ² , i ∈ {1,..., k}, is conjugate to an Anosov diffeomorphism of the torus T ².     

Free resolutions over short Gorenstein local rings

Let R be a local ring with maximal ideal ${\mathfrak{m}}$ admitting a non-zero element ${a\in\mathfrak{m}}$ for which the ideal (0 : a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M, with particular attention to the case when ${\mathfrak{m}^4=0}$ . Let e denote the minimal number of generators of ${\mathfrak{m}}$ . If R is Gorenstein with ${\mathfrak{m}^4=0}$ and e ?? 3, we show that ${{\rm P}_{M}^{R}(t)}$ is rational with denominator H _R (?t) =?1 ? et?+?et ² ? t ³, for each finitely generated R-module M. In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.     

Embeddings of Müntz Spaces: Composition Operators

Given a strictly increasing sequence ${\Lambda = (\lambda_n)}$ of nonnegative real numbers, with ${\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty}$ , the Müntz spaces ${M_\Lambda^p}$ are defined as the closure in L ^p ([0, 1]) of the monomials ${x^{\lambda_n}}$ . We discuss how properties of the embedding ${M_\Lambda^2\subset L^2(\mu)}$ , where?μ is a finite positive Borel measure on the interval [0, 1], have immediate consequences for composition operators on ${M^2_\Lambda}$ . We give criteria for composition operators to be bounded, compact, or to belong to the Schatten–von Neumann ideals.     

Traces of monotone Sobolev functions

In this paper we prove that ifu: ${\mathbb{B}}^n \to {\mathbb{R}}$ , where ${\mathbb{B}}^n $ is the unit ball in ? ⁿ , is a monotone function in the Sobolev space W^1·p ( ${\mathbb{B}}^n $ ), andn ? 1 <p ≤n, thenu has nontangential limits at all the points of $\partial {\mathbb{B}}^n $ except possibly on a set ofp-capacity zero. The key ingredient in the proof is an extension of a classical theorem of Lindelöf to monotone functions in W^1·p ( ${\mathbb{B}}^n $ ),n ? 1 <p ≤n.     

Lines Avoiding Balls in Three Dimensions Revisited

Let $\mathcal{B}$ be a collection of n arbitrary balls in ?³. We establish an almost-tight upper bound of O(n ^3+?? ), for any ??>0, on the complexity of the space $\mathcal{F}(\mathcal{B})$ of all the lines that avoid all the members of $\mathcal{B}$ . In particular, we prove that the balls of $\mathcal{B}$ admit O(n ^3+?? ) free isolated tangents, for any ??>0. This generalizes the result of Agarwal et al.?(Discrete Comput. Geom. 34:231?C250, 2005), who established this bound only for congruent balls, and solves an open problem posed in that paper. Our bound almost meets the recent lower bound of ??(n ³) of Glisse and Lazard (Proc. 26th Annu. Symp. Comput. Geom., pp. 48?C57, 2010). Our approach is constructive and yields an algorithm that computes the discrete representation of the boundary of $\mathcal{F}(B)$ in O(n ^3+?? ) time, for any ??>0.     

On Some Noteworthy Pairs of Ideals in Mod-R

An additive functor $F \colon {\mathcal A}\to{\mathcal B}$ between preadditive categories $\mathcal A$ and $\mathcal B$ is said to be a local functor if, for every morphism $f\colon A\to A'$ in $\mathcal A$ , F(f) isomorphism in $\mathcal B$ implies f isomorphism in $\mathcal A$ . We show that there exist several pairs $(\mathcal I_1,\mathcal I_2)$ of ideals of $\mathcal A$ for which the canonical functor $\mathcal A\to\mathcal A/\mathcal I_1\times \mathcal A/\mathcal I_2$ is a local functor. In most of our examples, the category $\mathcal A$ is a full subcategory of the category Mod?-R of all right modules over a ring R. These pairs of ideals arise in a surprisingly natural way and enjoy several properties. Ideals are kernels of functors, and most of our examples of ideals are kernels of important and well studied functors. E.g., (1) the kernel Δ of the canonical functor P of Mod?-R into its spectral category Spec(Mod?-R), so that Δ is the ideal of all morphisms with an essential kernel; (2) the kernel Σ of the dual functor F of P, so that Σ is the ideal of all morphisms with a superfluous image; (3) the kernels Δ⁽¹⁾ and Σ₍₁₎ of the first derived functors P ⁽¹⁾ and F ₍₁₎ of P and F, respectively; (4) the kernels of suitable functors Hom and ? and their first derived functors ${\rm Ext}^1_R$ and ${\rm Tor}^R_1$ .     

Mixed Invariant Subspaces over the Bidisk

It is known that the structure of invariant subspaces I of the Hardy space H ² over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces ${I\ominus zI}$ completely. In this paper, we study the structure of nontrivial closed subspaces N of H ² with ${T_zN\subset N}$ and ${T^*_wN\subset N}$ , which are called mixed invariant subspaces under T _z and ${T^*_w}$ . We know that the dimension of ${N\ominus zN}$ ranges from 1 to ??. If ${T^*_w(N\ominus zN)\subset N\ominus zN}$ , we may describe N completely. If ${T^*_w(N\ominus zN)\not\subset N\ominus zN}$ , it seems difficult to describe N generally. So we study N under the condition ${dim\,(N\ominus zN)=1}$ . Write ${M=H^2\ominus N}$ . We describe ${M\ominus wM}$ precisely. We give a characterization of N for which there is a nonzero function ${\varphi}$ in ${M\ominus wM}$ satisfying ${z^k\varphi\in M\ominus wM}$ for every k ?? 0. We also see that the space ${M\ominus wM}$ has a deep connection with the de Branges?CRovnyak spaces studied by Sarason.     

A group theoretic characterization of classical unitals

Let G be the group of projectivities stabilizing a unital $\mathcal{U}$ in PG(2,q ²). In?this paper, we prove that $\mathcal{U}$ is a classical unital if and only if there are two points in $\mathcal{U}$ such that the stabilizer of these two points in G has order?q ²?1.     

Isolated Hypersurface Singularities and Special Polynomial Realizations of Affine Quadrics

Let V, $\tilde{V}$ be hypersurface germs in ? ^m , each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for V, $\tilde{V}$ reduces to the linear equivalence problem for certain polynomials P, $\tilde{P}$ arising from the moduli algebras of V, $\tilde{V}$ . The polynomials P, $\tilde{P}$ are completely determined by their quadratic and cubic terms, hence the biholomorphic equivalence problem for V, $\tilde{V}$ in fact reduces to the linear equivalence problem for pairs of quadratic and cubic forms.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

Laguerre Isopararmetric Hypersurfaces in R^4

Let x : M → Rn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C , and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R4 .     

Infinite Combinatorial Issues Raised by Lifting Problems in Universal Algebra

The critical point between varieties? $\mathcal{A}$ and? $\mathcal{B}$ of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of? $\mathcal{A}$ but of no member of? $\mathcal{B}$ , if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (??,r,??)??m (for proving that a critical point is small). In the middle, we find ??-lifters of posets and the relation $(\kappa,{<}\lambda)\leadsto P$ , for infinite cardinals??? and??? and a poset?P.     

On the number of non-isomorphic subgraphs

Let $\mathcal{K}$ be the family of graphs on ω₁ without cliques or independent subsets of sizew ₁. We prove that

1. it is consistent with CH that everyGε $\mathcal{K}$ has 2^ω many pairwise non-isomorphic subgraphs,
2. the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω₁ either G?G[A] orG?G[B],
3. the failure of (*) is consistent with ZFC.

     

On functions whose symmetric part of gradient agree and a generalization of Reshetnyak’s compactness theorem

We consider the following question: Given a connected open domain ${\Omega \subset \mathbb{R}^n}$ , suppose ${u, v : \Omega \rightarrow \mathbb{R}^n}$ with det ${(\nabla u) > 0}$ , det ${(\nabla v) > 0}$ a.e. are such that ${\nabla u^T(x)\nabla u(x) = \nabla v(x)^T \nabla v(x)}$ a.e. , does this imply a global relation of the form ${\nabla v(x) = R\nabla u(x)}$ a.e. in Ω where ${R \in SO(n)}$ ? If u, v are C ¹ it is an exercise to see this true, if ${u, v\in W^{1,1}}$ we show this is false. In Theorem 1 we prove this question has a positive answer if ${v \in W^{1,1}}$ and ${u \in W^{1,n}}$ is a mapping of L ^p integrable dilatation for p > n ? 1. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville’s theorem that states that the differential inclusion ${\nabla u \in SO(n)}$ can only be satisfied by an affine mapping. Liouville’s corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence ${v_k \in W^{1,1}}$ for which $$\int \limits_{\Omega} {\rm dist}(\nabla v_k, SO(n))dz \rightarrow 0 \, {\rm as} \, k \rightarrow \infty.$$ Let S(·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 for any pair of weakly converging sequences ${v_k \in W^{1,p}}$ and ${u_k \in W^{1,\frac{p(n-1)}{p-1}}}$ (where ${p \in [1, n]}$ and the sequence (u _k ) has its dilatation pointwise bounded above by an L ^r integrable function, r > n ? 1) that satisfy ${\int_{\Omega} |S(\nabla u_k) - S(\nabla v_k)|^p dz \rightarrow 0}$ as k → ∞ and for which the sign of the det ${(\nabla v_k)}$ tends to 1 in L ¹. This result contains Reshetnyak’s theorem as the special case (u _k ) ≡ Id, p = 1.