Using properties of skew-Hamiltonian matrices and classic
connectedness results, we prove that the moduli space
$M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles
on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial
splitting on the general line, is smooth irreducible of
dimension $(r-2)n-\binom{r}{2}$ for $r=n$ and $n \ge 4$, and
$r=n-1$ and $n\ge 8$. We speculate that the result holds in
greater generality.

A category structure for Bratteli diagrams is proposed and a
functor from
the category of AF algebras to the category of Bratteli diagrams
is
constructed. Since isomorphism of Bratteli diagrams in this
category coincides
with Bratteli's notion of equivalence, we obtain in particular
a functorial formulation of Bratteli's
classification of AF algebras (and at the same time, of Glimm's
classification of UHF~algebras).
It is shown that the three approaches
to classification of AF~algebras, namely, through Bratteli diagrams,
K-theory, and
abstract classifying categories, are essentially the same
from a categorical point of view.

The symmetric group $\mathcal{S}_n$ acts on the power
set $\mathcal{P}(n)$ and also on the set of
square free polynomials in $n$ variables. These
two related representations are analyzed from the stability point
of view. An application is given for the action of the symmetric
group on the cohomology of the pure braid group.

In this paper, we mainly give a general explicit form of Cassels'
$p$-adic embedding theorem for number fields. We also give its
refined form in the case of cyclotomic fields. As a byproduct,
given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general
unconditional upper bound for the smallest prime number $p$ such
that $f$ has a simple root modulo $p$.

In this article we study exponential trichotomy for infinite dimensional
discrete time dynamical systems. The goal of this article is to prove that
finite time exponential trichotomy conditions allow to derive exponential
trichotomy for any times. We present an application to the case of pseudo
orbits in some neighborhood of a normally hyperbolic set.

Let $\mathbf{TB}$ be the category of totally bounded, locally
compact metric spaces
with the $C_0$ coarse structures. We show that if $X$ and $Y$
are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent
if and only if their Higson coronas are homeomorphic. In fact,
the Higson corona functor gives an equivalence of categories
$\mathbf{TB}\to\mathbf{K}$, where $\mathbf{K}$ is the category
of compact metrizable spaces. We use this fact to show that the
continuously controlled coarse structure on a locally compact
space $X$ induced by some metrizable compactification $\tilde{X}$
is determined only by the topology of the remainder $\tilde{X}\setminus
X$.

We show that the moduli space
of parabolic bundles on the projective line
and the polygon space are isomorphic,
both as complex manifolds
and symplectic manifolds equipped with structures of completely integrable systems,
if the stability parameters are
small.

We extend the classicial notion of an outer action
$\alpha$ of a group $G$ on a unital ring $A$
to the case when $\alpha$ is a partial action
on ideals, all of which have local units.
We show that if $\alpha$ is an outer partial
action of an abelian group $G$,
then its associated partial skew group
ring $A \star_\alpha G$ is simple if and only if
$A$ is $G$-simple.
This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights
or in the class of $QC(\mathbb{R}^n)$ weights, and
$L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$
the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$,
$n\ge 2$. In this article, the authors establish the non-tangential
maximal function characterization
of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for
$p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and
$w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$,
the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$
is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical
Hardy space $H_w^p(\mathbb{R}^n)$.