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961  Multiplicative Isometries and Isometric ZeroDivisors Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
For some Banach spaces of analytic functions in the unit disk
(weighted Bergman spaces, Bloch space, Dirichlettype spaces), the
isometric pointwise multipliers are found to be unimodular constants.
As a consequence, it is shown that none of those spaces have isometric
zerodivisors. Isometric coefficient multipliers are also
investigated.


975  Revisiting TietzeNakajima: Local and Global Convexity for Maps Bjorndahl, Christina; Karshon, Yael
A theorem of Tietze and Nakajima, from 1928, asserts that
if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex,
then it is convex.
We give an analogous ``local to global convexity" theorem
when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map
from a topological space $X$ to $\mathbb{R}^n$ that satisfies
certain local properties.
Our motivation comes from the CondevauxDazordMolino proof
of the AtiyahGuilleminSternberg convexity theorem in symplectic geometry.


994  Curvature Bounds for Surfaces in Hyperbolic 3Manifolds Breslin, William
A triangulation of a hyperbolic $3$manifold is Lthick if each
tetrahedron having all vertices in the thick part of $M$ is
$L$bilipschitz diffeomorphic to the standard Euclidean tetrahedron.
We show that there exists a fixed constant $L$ such that every
complete hyperbolic $3$manifold has an $L$thick geodesic
triangulation. We use this to prove the existence of universal bounds on
the principal curvatures of $\pi_1$injective surfaces and strongly
irreducible Heegaard surfaces in hyperbolic $3$manifolds.


1011  Functoriality of the Canonical Fractional Galois Ideal Buckingham, Paul; Snaith, Victor
The fractional Galois ideal
is a conjectural improvement on the higher Stickelberger
ideals defined at negative integers, and is expected to provide
nontrivial annihilators for higher $K$groups of rings of integers of
number fields. In this article, we extend the definition of the
fractional Galois ideal to arbitrary (possibly infinite and
nonabelian) Galois extensions of number fields under the assumption
of Stark's conjectures and prove naturality properties under
canonical changes of extension. We discuss applications of this to the
construction of ideals in noncommutative Iwasawa algebras.


1037  Riemann Extensions of TorsionFree Connections with Degenerate Ricci Tensor CalviñoLouzao, E.; GarcíaRío, E.; VázquezLorenzo, R.
{Correspondence} between torsionfree connections with {nilpotent skewsymmetric curvature operator} and IP Riemann
extensions is shown. Some consequences are derived in the study of
fourdimensional IP metrics and locally homogeneous affine surfaces.


1058  On a Conjecture of S. Stahl Chen, Yichao; Liu, Yanpei
S. Stahl conjectured that the zeros of genus polynomials are real. In
this note, we disprove this conjecture.


1060  Heegner Points over Towers of Kummer Extensions Darmon, Henri; Tian, Ye
Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension
generated by a primitive $p^n$th root of unity and a $p^n$th root of
$a$ for a fixed $a\in \mathbb{Q}^\times\{\pm 1\}$. A detailed case study
by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these
authors to predict unbounded and strikingly regular growth for the
rank of $E$ over $L_n$ in certain cases. The aim of this note is to
explain how some of these predictions might be accounted for by
Heegner points arising from a varying collection of Shimura curve
parametrisations.


1082  The Fundamental Group of $S^1$manifolds Godinho, Leonor; SousaDias, M. E.
We address the problem of computing the fundamental
group of a symplectic $S^1$manifold for nonHamiltonian actions on
compact manifolds, and for Hamiltonian actions on noncompact
manifolds with a proper moment map. We generalize known results for
compact manifolds equipped with a Hamiltonian $S^1$action. Several
examples are presented to illustrate our main results.


1099  Character Sums to Smooth Moduli are Small Goldmakher, Leo
Recently, Granville and Soundararajan have made
fundamental breakthroughs in the study of character sums. Building
on their work and using estimates on short character sums developed
by GrahamRingrose and Iwaniec, we improve the
PólyaVinogradov inequality for characters with smooth conductor.


1116  Degenerate pLaplacian Operators and Hardy Type Inequalities on
HType Groups Jin, Yongyang; Zhang, Genkai
Let $\mathbb G$ be a steptwo nilpotent group of Htype with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$Laplacian operator $L_{p,k} u= \operatorname{div}_X (\nabla_{X} u^{p2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.


1131  Moduli Spaces of Reflexive Sheaves of Rank 2 Kleppe, Jan O.
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
e.g., the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(e.g., of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.


1155  Moments of the Critical Values of Families of Elliptic Curves, with Applications Young, Matthew P.
We make conjectures on the moments of the central values of the family
of all elliptic curves and on the moments of the first derivative of
the central values of a large family of positive rank curves. In both
cases the order of magnitude is the same as that of the moments of the
central values of an orthogonal family of $L$functions. Notably, we
predict that the critical values of all rank $1$ elliptic curves is
logarithmically larger than the rank $1$ curves in the positive rank
family.


1182  A Fractal Function Related to the JohnNirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$ Yue, Hong
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces
is defined as a Haar wavelet decomposition, with the coefficients
depending on a fixed parameter $\beta>0$. On its support $I_0=[0,
1]^n$, $f(x)$ can be expressed by the binary expansions of the
coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb
R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for
$\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a
JohnNirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a selfaffine function. It is continuous almost
everywhere and discontinuous at all dyadic points inside $I_0$. In
addition, it is not monotone along any coordinate direction in any
small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from
$I_0$ to $[\frac{1}{12^{\beta}}, \frac{1}{12^{\beta}}]$, and the
graph of $f$ has a noninteger fractal dimension $n+1\beta$.

