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3  On Hilbert Covariants Abdesselam, Abdelmalek; Chipalkatti, Jaydeep
Let $F$ denote a binary form of order $d$ over the
complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant
$\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an
order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give
defining equations for the image variety $X$ of an embedding $\mathbf{P}^r
\hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of
the Hilbert covariant; and simultaneously situate it into a wider class of
covariants called the Göttingen covariants, all of which vanish on
$X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$
defines $X$ as a scheme. Finally, we exhibit a generalisation of the
Göttingen covariants to $n$ary forms using the classical Clebsch transfer principle.


31  Symplectic Foliations and Generalized Complex Structures Bailey, Michael
We answer the natural question: when is a transversely holomorphic
symplectic foliation induced by a generalized complex structure? The
leafwise symplectic form and transverse complex structure determine an
obstruction class in a certain cohomology, which vanishes if and only
if our question has an affirmative answer. We first study a component
of this obstruction, which gives the condition that the leafwise
cohomology class of the symplectic form must be transversely
pluriharmonic. As a consequence, under certain topological
hypotheses, we infer that we actually have a symplectic fibre bundle
over a complex base. We then show how to compute the full obstruction
via a spectral sequence. We give various concrete necessary and
sufficient conditions for the vanishing of the obstruction.
Throughout, we give examples to test the sharpness of these
conditions, including a symplectic fibre bundle over a complex base
which does not come from a generalized complex structure, and a
regular generalized complex structure which is very unlike a
symplectic fibre bundle, i.e., for which nearby leaves are not
symplectomorphic.


57  Perfect Orderings on Finite Rank Bratteli Diagrams Bezuglyi, S.; Kwiatkowski, J.; Yassawi, R.
Given a Bratteli diagram $B$, we study the set $\mathcal O_B$ of all
possible orderings on $B$ and its subset
$\mathcal P_B$ consisting of perfect orderings that produce
BratteliVershik topological dynamical systems (Vershik maps). We
give necessary and sufficient conditions for the ordering $\omega$ to be
perfect. On the other hand, a
wide class of nonsimple Bratteli diagrams that do not admit Vershik
maps is explicitly described. In the case of finite rank Bratteli
diagrams, we show that the existence of perfect orderings with a prescribed
number of extreme paths constrains significantly the values of the entries of
the incidence matrices and the structure of the diagram $B$. Our
proofs are based on the new notions of skeletons and
associated graphs, defined and studied in the paper. For a Bratteli
diagram $B$ of rank $k$, we endow the set $\mathcal O_B$ with product
measure $\mu$ and prove that there is some $1 \leq j\leq k$ such that
$\mu$almost all orderings on $B$ have $j$ maximal and $j$ minimal
paths. If $j$ is strictly greater than the number of minimal
components that $B$ has, then $\mu$almost all orderings are imperfect.


102  Continuity of convolution of test functions on Lie groups Birth, Lidia; Glöckner, Helge
For a Lie group $G$, we show that the map
$C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$,
$(\gamma,\eta)\mapsto \gamma*\eta$
taking a pair of
test functions to their convolution is continuous if and only if $G$ is $\sigma$compact.
More generally, consider $r,s,t
\in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$
and a continuous bilinear map $b\colon E_1\times E_2\to F$
to a complete locally convex space $F$.
Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$,
$(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map.
The main result is a characterization of those $(G,r,s,t,b)$
for which $\beta$ is continuous.
Convolution
of compactly supported continuous functions on a locally compact group
is also discussed, as well as convolution of compactly supported $L^1$functions
and convolution of compactly supported Radon measures.


141  Existence of Taut Foliations on Seifert Fibered Homology $3$spheres CaillatGibert, Shanti; Matignon, Daniel
This paper concerns the problem of existence of taut foliations among $3$manifolds.
Since the contribution of David Gabai,
we know that closed $3$manifolds with nontrivial second homology group
admit a taut foliation.
The essential part of this paper focuses on Seifert fibered homology $3$spheres.
The result is quite different if they are integral or rational but nonintegral homology $3$spheres.
Concerning integral homology $3$spheres, we can see that all but the $3$sphere and the Poincaré $3$sphere admit a taut foliation.
Concerning nonintegral homology $3$spheres,
we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation.
Moreover, we show that the geometries do not determine the existence of taut foliations
on nonintegral Seifert fibered homology $3$spheres.


170  Modular Abelian Varieties Over Number Fields Guitart, Xavier; Quer, Jordi
The main result of this paper is a characterization of the abelian
varieties $B/K$ defined over Galois number fields with the
property that the $L$function $L(B/K;s)$ is a product of
$L$functions of nonCM newforms over $\mathbb Q$ for congruence
subgroups of the form $\Gamma_1(N)$. The characterization involves the
structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of
$B$, and a Galois cohomology class attached to $B/K$.


197  On Hyperbolicity of Domains with Strictly Pseudoconvex Ends Harris, Adam; Kolář, Martin
This article establishes a sufficient condition for Kobayashi
hyperbolicity of unbounded domains in terms of curvature.
Specifically, when $\Omega\subset{\mathbb C}^{n}$ corresponds to a
sublevel set of a smooth, realvalued function $\Psi$, such that the
form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is Kähler and
has bounded curvature outside a bounded subset, then this domain
admits a hermitian metric of strictly negative holomorphic sectional
curvature.


205  Generalized Frobenius Algebras and Hopf Algebras Iovanov, Miodrag Cristian
"CoFrobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
leftright symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasicoFrobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the
sense that certain coproduct or product powers of these objects are
isomorphic. Fundamental results of Hopf algebras, such as the
equivalent characterizations of Hopf algebras with nonzero integrals
as left (or right) coFrobenius, QcF, semiperfect or with nonzero
rational dual, as well as the uniqueness of integrals and a short
proof of the bijectivity of the antipode for such Hopf algebras all
follow as a consequence of these results. This gives a purely
representation theoretic approach to many of the basic fundamental
results in the theory of Hopf algebras. Furthermore, we introduce a
general concept of Frobenius algebra, which makes sense for infinite
dimensional and for topological algebras, and specializes to the
classical notion in the finite case. This will be a topological
algebra $A$ that is isomorphic to its complete topological dual
$A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only
if $A$ is the dual $C^*$ of a (quasi)coFrobenius coalgebra $C$. We
give many examples of coFrobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
qhomological algebra, topology or graph theory, showing the
importance of the concept.
