We determine which groups ${\mathbb{Z}}/M{\mathbb{Z}}\oplus{\mathbb{Z}}/N{\mathbb{Z}}$ occur infinitely often as torsion groups $E(K)_{\operatorname{tors}}$ when $K$ varies over all quartic number fields and $E$ varies over all elliptic curves over $K$.

We prove a lifting theorem (from any characteristic $p$ to characteristic zero) for holomorphic Picard modular forms of any weight and level. The proof makes use of the knowledge of the geometry of the toroidal compactification of the Picard modular surface.

This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space $A^{1}(M)$, for such a Riemann surface $M$, is isomorphic to the Banach space of summable sequence, $l^{1}$. This implies that whenever $M$ and $N$ are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then $A^{1}(M)$ is isomorphic to $A^{1}(N)$. It is known from V. Markovic that if there is a linear isometry between $A^{1}(M)$ and $A^{1}(N)$, for two Riemann surfaces $M$ and $N$ of non-exceptional type, then this isometry is induced by a conformal mapping between $M$ and $N$. As a corollary to this rigidity theorem presented here, taking the Banach duals of $A^{1}(M)$ and $l^{1}$ shows that the space of holomorphic quadratic differentials on $M,\ Q(M)$, is isomorphic to the Banach space of bounded sequences, $l^{\infty }$. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent.

We consider the classical universal covering $\exp: {{\mathbb C}}\To {{\mathbb C}}^*$ of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion-free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of ${{\mathbb C}}$. This can also be seen as a model-theoretic statement on the categoricity of a corresponding $L_{\omega_1,\omega}$-sentence. The proof is a combination of arithmetic and model-theoretic methods.

Let $Y$ be a reduced irreducible projective curve of arithmetic genus $g\ge 2$ with at most ordinary nodes as singularities. For a subsheaf $F$ of rank $r'$, degree $d'$ of a torsion-free sheaf $E$ of rank $r$, degree $d$, let $s(E,F) = r'd-rd'$. Define $s_{r'}(E) = {\rm min} s(E,F)$, where the minimum is taken over all subsheaves of $E$ of rank $r'$. For a fixed $r', s_{r'}$ defines a stratification of the moduli space $U(r,d)$ of stable torsion-free sheaves of rank $r$, degree $d$ by locally closed subsets $U_{r',s}$. We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank $r'$ for $s\le r'(r-r')(g-1)$ (respectively $s<r'(r-r')(g-1)$). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve $Y$ is nonspecial.

We revise the concept of compact tripotent in the bidual space of a JB*-triple. This concept was introduced by Edwards and Rüttimann generalizing the ideas developed by Akemann for compact projections in the bidual of a C*-algebra. We also obtain some characterizations of weak compactness in the dual space of a JC*-triple, showing that a bounded subset in the dual space of a JC*-triple is relatively weakly compact if and only if its restriction to any abelian maximal subtriple $C$ is relatively weakly compact in the dual of $C$. This generalizes a very useful result by Pfitzner in the setting of C*-algebras. As a consequence we obtain a Dieudonné theorem for JC*-triples which generalizes the one obtained by Brooks, Saitô and Wright for C*-algebras.

On a large class of post-critically finite (finitely ramified) self-similar fractals with possibly little symmetry, we consider the question of existence and uniqueness of a Laplace operator. By considering positive refinement weights (local scaling factors) which are not necessarily equal, we show that for each such fractal, under a certain condition, there are corresponding refinement weights which support a unique self-similar Dirichlet form. As compared with previous results, our technique allows us to replace symmetry by connectivity arguments.

We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by Lions in 1962. To do this we develop a general theory of quadratic estimates and functional calculi for complex perturbations of Dirac-type operators on Lipschitz domains.

Szemerédi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.

In 1868 Zolotarev determined the polynomial which deviates least from zero with respect to the maximum norm on $[-1,1]$ among all polynomials of the form $x^n + \sigma x^{n-1} + a_{n-2}x^{n-2} + \ldots +a_1x + a_0$, where $\sigma \in {\mathbb R}$ is given. The polynomial was given explicitly in terms of elliptic functions by Zolotarev. It is now called the Zolotarev polynomial. Zolotarev also gave an explicit expression for the minimum deviation. In the sequel attempts have been made to replace the elliptic functions and to express the Zolotarev polynomial and the minimum deviation in terms of elementary functions, at least asymptotically. In 1913 Bernstein succeeded in finding an asymptotic formula for the minimum deviation, which has been improved several times since then. Here we give the first asymptotic representation of the Zolotarev polynomials. For the asymptotic representation we use the rational functions introduced by Bernstein. Furthermore, we obtain asymptotic representations of minimal polynomials with interpolation constraints which are of interest in the theory of the iterative solution of inconsistent linear systems of equations.

We introduce a numerical radius operator space $(X, \mathcal{W}_n)$. The conditions to be a numerical radius operator space are weaker than Ruan's axiom for an operator space $(X, \mathcal{O}_n)$. Let $w(\cdot)$ be the numerical radius on $\mathbb{B}(\mathcal{H})$. It is shown that, if $X$ admits a norm $\mathcal{W}_n(\cdot)$ on the matrix space $\mathbb{M}_n(X)$ which satisfies the conditions, then there is a complete isometry, in the sense of the norms $\mathcal{W}_n(\cdot)$ and $w_n(\cdot)$, from $(X, \mathcal{W}_n)$ into $(\mathbb{B}(\mathcal{H}), w_n)$. We study the relationship between the operator space $(X, \mathcal{O}_n)$ and the numerical radius operator space $(X, \mathcal{W}_n)$. The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.

We show that, if $A$ is a separable simple unital $C^{*}$-algebra that absorbs the Jiang–Su algebra ${{\mathcal Z}}$ tensorially and that has real rank zero and finite decomposition rank, then $A$ is tracially approximately finite-dimensional in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of $C^{*}$-algebras as above that, additionally, satisfy the universal coefficients theorem. In particular, such algebras are completely determined by their ordered $K$-theory. They are approximately homogeneous of topological dimension less than or equal to three, approximately subhomogeneous of topological dimension at most two and their decomposition rank also is no greater than two.

In the existing theory of self-affine tiles, one knows that the Lebesgue measure of any integral self-affine tile corresponding to a standard digit set must be a positive integer and every integral self-affine tile admits some lattice $\varGamma\subseteq\mathbb{Z}^n$ as a translation tiling set of $\mathbb{R}^n$. In this paper, we give algorithms to evaluate the Lebesgue measure of any such integral self-affine tile $K$ and to determine all of the lattice tilings of $\mathbb^n$ by $K$. Moreover, we also propose and determine algorithmically another type of translation tiling of $\mathbb{R}^n$ by $K$, which we call natural tiling. We also provide an algorithm to decide whether or not Lebesgue measure of the set $K\cap (K+j),\ j\in\mathbb{Z}^n$, is strictly positive.

In this paper we study the metric geometry of the space $\Sigma$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $\tau$. The trace induces an incomplete Riemannian metric $\langle x,y\rangle_a=\tau (ya^{-1}xa^{-1})$, and, though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance, we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds $M$ of $\Sigma$; and under a suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained via a nonlinear orthogonal projection $\Pi_M:\Sigma\to M$, a map which turns out to be contractive for the geodesic distance.

This article introduces the notion of 2-ruled 4-folds: submanifolds of ${\mathbb{R}}^n$ fibred over a 2-fold $\Sigma$ by affine 2-planes. This is motivated by a paper by Joyce and previous work of the present author. A 2-ruled 4-fold $M$ is r-framed if an oriented basis is smoothly assigned to each fibre, and then we may write $M$ in terms of orthogonal smooth maps $\phi_1,\phi_2:\Sigma\rightarrow\mathcal{S}^{n-1}$ and a smooth map $\psi:\Sigma\rightarrow{\mathbb{R}}^n$. We focus on 2-ruled Cayley 4-folds in ${\mathbb{R}}^8$ as certain other calibrated 4-folds in ${\mathbb{R}}^7$ and ${\mathbb{R}}^8$ can be considered as special cases. The main result characterizes non-planar, r-framed, 2-ruled Cayley 4-folds, using a coupled system of nonlinear, first-order, partial differential equations that $\phi_1$ and $\phi_2$ satisfy, and another such equation on $\psi$ which is linear in $\psi$. We give a means of constructing 2-ruled Cayley 4-folds starting from particular 2-ruled Cayley cones using holomorphic vector fields. This is used to give explicit examples of ${\mathbin{\rm U}}(1)$-invariant 2-ruled Cayley 4-folds asymptotic to a ${\mathbin{\rm U}}(1)^3$-invariant 2-ruled Cayley cone. Examples are also given based on ruled calibrated 3-folds in ${\mathbb{C}}^3$ and ${\mathbb{R}}^7$ and complex cones in ${\mathbb{C}}^4$.

In this paper we consider an aperiodic integer-valued random walk $S$ and a process $S^{\ast }$ that is a harmonic transform of $S$ killed when it first enters the negative half; informally, $S^{\ast }$ is ‘$S$ conditioned to stay non-negative’. If $S$ is in the domain of attraction of the standard normal law, without centring, a suitably normed and linearly interpolated version of $S$ converges weakly to standard Brownian motion, and our main result is that under the same assumptions a corresponding statement holds for $S^{\ast }$, the limit of course being the three-dimensional Bessel process. As this process can be thought of as Brownian motion conditioned to stay non-negative, in essence our result shows that the interchange of the two limit operations is valid. We also establish some related results, including a local limit theorem for $S^{\ast }$, and a bivariate renewal theorem for the ladder time and height process, which may be of independent interest.

We prove that every limit group acts geometrically on a CAT(0) space with the isolated flats property.