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481  Number of Right Ideals and a $q$analogue of Indecomposable Permutations Bacher, Roland; Reutenauer, Christophe
We prove that the number of right ideals of codimension $n$ in
the algebra of noncommutative Laurent polynomials in two variables over the finite field $\mathbb F_q$ is equal to
$(q1)^{n+1} q^{\frac{(n+1)(n2)}{2}}\sum_\theta q^{inv(\theta)}$,
where the
sum is over all indecomposable permutations in $S_{n+1}$ and
where $inv(\theta)$
stands for the number of inversions of $\theta$.


504  Integrable Systems and Torelli Theorems for the Moduli Spaces of Parabolic Bundles and Parabolic Higgs Bundles Biswas, Indranil; Gómez, Tomás L.; Logares, Marina
We prove a Torelli theorem for the moduli space of semistable
parabolic Higgs bundles over a smooth complex projective algebraic
curve under the assumption that the parabolic weight system is
generic. When the genus is at least two, using this result
we also
prove a Torelli theorem for the moduli space of semistable
parabolic
bundles of rank at least two with generic parabolic weights.
The key
input in the proofs is a method of
J.C. Hurtubise.


521  Optimization Related to Some Nonlocal Problems of Kirchhoff Type Emamizadeh, Behrouz; Farjudian, Amin; ZivariRezapour, Mohsen
In this paper we introduce two rearrangement optimization
problems, one being a maximization and the other a minimization
problem, related to a nonlocal boundary value problem of Kirchhoff
type. Using the theory of rearrangements as developed by
G. R. Burton we are able to show that both problems are solvable,
and derive the corresponding optimality conditions. These conditions
in turn provide information concerning the locations of the
optimal
solutions. The strict convexity of the energy functional plays
a
crucial role in both problems. The popular case in which the
rearrangement class (i.e., the admissible set) is generated
by a
characteristic function is also considered. We show that in
this
case, the maximization problem gives rise to a free boundary
problem
of obstacle type, which turns out to be unstable. On the other
hand,
the minimization problem leads to another free boundary problem
of
obstacle type, which is stable. Some numerical results are
included
to confirm the theory.


541  Strongly Incompressible Curves GarciaArmas, Mario
Let $G$ be a finite group. A faithful $G$variety $X$ is called
strongly incompressible if every dominant $G$equivariant rational
map of $X$ onto another faithful $G$variety $Y$ is birational.
We settle the problem of existence of strongly incompressible
$G$curves for any finite group $G$ and any base field $k$ of
characteristic zero.


571  Les $\theta$régulateurs locaux d'un nombre algébrique : Conjectures $p$adiques Gras, Georges
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that
$\operatorname{Reg}_\infty (\eta) \ne 0$.
We define the local $\theta$regulators $\Delta_p^\theta(\eta)
\in \mathbb{F}_p$
for the $\mathbb{Q}_p\,$irreducible characters $\theta$ of
$G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \,
V_\theta$ is associated with
$\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta
\geq 1$.
Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$
modulo $p$ in the factorization
$\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of
$\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]}
}$ (normalized $p$adic regulator).
From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal
L}^\theta \simeq \delta \, V_\theta\big )
\leq p^{ f \delta^2}$ ($f \geq 1$ is a residue degree) and the
BorelCantelli heuristic,
we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$
is a $p$adic unit or that
$p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$
with $f=\delta=1$); this obstruction may be lifted assuming the
existence of a binomial probability law
confirmed through numerical studies
(groups $C_3$, $C_5$, $D_6$).
This conjecture would imply that, for all $p$ large enough,
Fermat quotients, normalized $p$adic
regulators are $p$adic units and that
number fields are $p$rational.
We recall some deep cohomological results that
may strengthen such conjectures.


625  Rigidity and Height Bounds for Certain Postcritically Finite Endomorphisms of $\mathbb P^N$ Ingram, Patrick
The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called postcritically finite
(PCF) if the forward image of the critical locus, under iteration
of $f$, has algebraic support. In the case $N=1$, a result of
Thurston implies that there are no algebraic families of PCF
morphisms, other than a wellunderstood exceptional class known
as the flexible Lattès maps. A related arithmetic result
states that the set of PCF morphisms corresponds to a set of
bounded height in the moduli space of univariate rational functions.
We prove corresponding results for a certain subclass of the
regular polynomial endomorphisms of $\mathbb{P}^N$, for any $N$.


655  Discrete Curvature and Abelian Groups Klartag, Bo'az; Kozma, Gady; Ralli, Peter; Tetali, Prasad
We study a natural discrete Bochnertype inequality on graphs,
and explore its merit as a notion of ``curvature'' in discrete
spaces.
An appealing feature of this discrete version of the socalled
$\Gamma_2$calculus (of BakryÉmery) seems to be that it is
fairly
straightforward to compute this notion of curvature parameter
for
several specific graphs of interest  particularly, abelian
groups, slices of the hypercube, and the symmetric group under
various sets of generators.
We further develop this notion by deriving Busertype inequalities
(à la Ledoux), relating functional and isoperimetric constants
associated with a graph.
Our derivations provide a tight bound on the Cheeger constant
(i.e., the edgeisoperimetric constant) in terms of
the spectral gap, for graphs with nonnegative curvature, particularly,
the class of abelian Cayley graphs  a result of independent
interest.


675  Monotone Classes of Dendrites MartínezdelaVega, Veronica; Mouron, Christopher
Continua $X$ and $Y$ are monotone equivalent
if there exist monotone onto maps $f:X\longrightarrow Y$ and
$g:Y\longrightarrow X$. A continuum $X$ is isolated with respect
to monotone maps if every continuum that is monotone equivalent
to $X$ must also be homeomorphic to
$X$. In this paper we show that a dendrite $X$ is isolated with
respect to
monotone maps if and only if the set of ramification points of
$X$ is
finite. In this way we fully characterize the classes of dendrites
that are
monotone isolated.


698  Quantum Families of Invertible Maps and Related Problems Skalski, Adam; Sołtan, Piotr
The notion of families of quantum invertible maps (C$^*$algebra
homomorphisms satisfying Podleś' condition) is employed to strengthen
and reinterpret several results concerning universal quantum
groups acting on finite quantum spaces. In particular Wang's
quantum automorphism groups are shown to be universal with respect
to quantum families of invertible maps. Further the construction
of the Hopf image of Banica and Bichon is phrased in the purely
analytic language and employed to define the quantum subgroup
generated by a family of quantum subgroups or more generally
a family of quantum invertible maps.

