For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in
K}$ which is invariant, in law, under unitary conjugation, we
give general sufficient conditions for central limit theorems
for random variables of the type $\operatorname{Tr}(\mathbf{A}_k
\mathbf{M})$, where the matrix $\mathbf{M}$ is deterministic
(such random variables include for example the normalized matrix
entries of the $\mathbf{A}_k$'s). A consequence is the asymptotic
independence of the projection of the matrices $\mathbf{A}_k$
onto the subspace of null trace matrices from their projections
onto the orthogonal of this subspace. These results are used
to study the asymptotic behavior of the outliers of a spiked
elliptic random matrix. More precisely, we show that the fluctuations
of these outliers around their limits can have various rates
of convergence, depending on the Jordan Canonical Form of the
additive perturbation. Also, some correlations can arise between
outliers at a macroscopic distance from each other. These phenomena
have already been observed
with random matrices
from the Single Ring Theorem.
We study comparison properties in the category $\mathrm{Cu}$ aiming to
lift results to the C*-algebraic setting. We introduce a new
comparison property and relate it to both the CFP and $\omega$-comparison.
We show differences of all properties by providing examples,
which suggest that the corona factorization for C*-algebras might
allow for both finite and infinite projections. In addition,
we show that R{\o}rdam's simple, nuclear C*-algebra with a finite
and an infinite projection does not have the CFP.
We study the Bishop-Phelps-Bollobás property (BPBp for short)
for compact operators. We present some abstract techniques which
allows to carry the BPBp for compact operators from sequence
spaces to function spaces. As main applications, we prove the
following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$
has the BPBp for compact operators, then so do $(C_0(L),Y)$ for
every locally compact Hausdorff topological space $L$ and $(X,Y)$
whenever $X^*$ is isometrically isomorphic to $\ell_1$.
If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$
has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$
for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$
has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional
or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any
positive measure $\nu$ and $1\lt p\lt \infty$.
For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact
operators, then so does $(X,L_p(\mu,Y))$ for every positive measure
$\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$
has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$
for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$
for every compact Hausdorff topological space $K$.
This note provides a correct proof of the result claimed by the
second author that locally compact normal spaces are collectionwise
Hausdorff in certain models obtained by forcing with a coherent
Souslin tree. A novel feature of the proof is the use of saturation
of the non-stationary ideal on $\omega_1$, as well as of a strong
form of Chang's Conjecture. Together with other improvements,
this enables the consistent characterization of locally compact
hereditarily paracompact spaces as those locally compact, hereditarily
normal spaces that do not include a copy of $\omega_1$.
This paper introduces a class of abstract linear representations
on
Banach convolution function algebras over
homogeneous spaces of compact groups. Let $G$ be a compact group
and $H$ be a closed subgroup of $G$.
Let $\mu$ be the normalized $G$-invariant measure over the compact
homogeneous space $G/H$ associated to the
Weil's formula and $1\le p\lt \infty$.
We then present a structured class of abstract linear representations
of the
Banach convolution function algebras $L^p(G/H,\mu)$.
Let $\mathbf{F}_{q}[T]$ be the ring of polynomials over the finite
field of $q$ elements, and $Y$ be a large integer. We say a polynomial
in $\mathbf{F}_{q}[T]$ is $Y$-smooth if all of its irreducible
factors
are of degree at most $Y$. We show that a ternary additive equation
$a+b=c$ over $Y$-smooth polynomials has many solutions. As an
application,
if $S$ is the set of first $s$ primes in $\mathbf{F}_{q}[T]$ and
$s$ is large, we prove that the $S$-unit equation $u+v=1$ has at
least $\exp(s^{1/6-\epsilon}\log q)$ solutions.
For a finite abelian $p$-group $A$ of rank $d=\dim A/pA$, let
$\mathbb{M}_A := \log_p |A|^{1/d}$ be its
(logarithmic) mean exponent. We study the behavior of
the mean exponent of $p$-class groups in pro-$p$ towers $\mathrm{L}/K$
of number fields. Via a combination of results from analytic
and algebraic number theory, we construct infinite tamely
ramified pro-$p$ towers in which the mean exponent of $p$-class
groups remains bounded. Several explicit
examples are given with $p=2$. Turning to group theory, we
introduce an invariant $\underline{\mathbb{M}}(G)$ attached to a finitely generated
pro-$p$ group $G$; when $G=\operatorname{Gal}(\mathrm{L}/\mathrm{K})$, where $\mathrm{L}$ is the Hilbert
$p$-class field tower of a number field $K$, $\underline{\mathbb{M}}(G)$ measures
the asymptotic behavior of the mean exponent of $p$-class groups
inside $\mathrm{L}/\mathrm{K}$. We compare and contrast the behavior of this
invariant in analytic versus non-analytic groups. We exploit
the interplay of group-theoretical and number-theoretical perspectives
on this invariant and explore some open questions that arise
as a result, which may be of independent interest in group theory.
We study a second-order ordinary differential
equation coming from the Kepler problem on $\mathbb{S}^2$. The
forcing term under consideration is a piecewise constant with
singular nonlinearity which changes sign. We establish necessary
and
sufficient conditions to the existence and multiplicity of
$T$-periodic solutions.
The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}\mathbb{P}^n$ are computed
along with
the Atiyah-Hirzebruch spectral sequence for
$ER(2)^*(\mathbb{C}\mathbb{P}^\infty)$.
This, along with other papers in this series, gives
us the $ER(2)$-cohomology of all Eilenberg-MacLane spaces.
I construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty$
of real functions with logarithmic generalized power series as
asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation
and $\log$-composition; in particular, $\mathcal{F}$ is a Hardy field.
Moreover, the field $\mathcal{F} \circ (-\log)$ of germs at $0^+$ contains
all transition maps of hyperbolic saddles of planar real analytic
vector fields.
Let $\Theta ^{[j]}$ be an analogue of the Ramanujan theta operator
for Siegel modular forms.
For a given prime $p$, we give the weights of elements of mod
$p$ kernel of $\Theta ^{[j]}$,
where the mod $p$ kernel of $\Theta ^{[j]}$ is the set of all
Siegel modular forms $F$ such that $\Theta ^{[j]}(F)$ is congruent
to zero modulo $p$.
In order to construct examples of the mod $p$ kernel of $\Theta
^{[j]}$ from any Siegel modular form,
we introduce new operators $A^{(j)}(M)$ and show the modularity
of $F|A^{(j)}(M)$ when $F$ is a Siegel modular form.
Finally, we give some examples of the mod $p$ kernel of $\Theta
^{[j]}$ and the filtrations of some of them.
Let $W$ be a compact simply connected triangulated manifold with
boundary and $K\subset W$ be a subpolyhedron. We construct an
algebraic model of the rational homotopy type of $W\backslash K$ out of
a model of the map of pairs $(K,K \cap \partial W)\hookrightarrow
(W,\partial W)$ under some high codimension hypothesis.
We deduce the rational homotopy invariance of the configuration
space
of two points in a compact manifold with boundary under 2-connectedness
hypotheses. Also, we exhibit
nice explicit models of these configuration spaces for a large
class
of compact manifolds.
We address the classification problem for graph $C^*$-algebras of
finite graphs (finitely many edges and vertices), containing
the class of Cuntz-Krieger algebras as a
prominent special case. Contrasting earlier work, we do not assume
that the graphs satisfy the standard condition (K), so that the
graph
$C^*$-algebras may come with uncountably many ideals.
We find that in this generality, stable isomorphism of graph
$C^*$-algebras does not coincide with the geometric notion of Cuntz
move equivalence. However, adding a modest condition on the
graphs, the two notions are proved to be mutually equivalent and
equivalent to the $C^*$-algebras having isomorphic $K$-theories. This
proves in turn that under this condition, the graph
$C^*$-algebras are in fact classifiable by $K$-theory, providing in
particular complete classification when the $C^*$-algebras in question
are either of real rank zero or type I/postliminal. The key ingredient
in obtaining these results is a characterization of Cuntz move
equivalence using the adjacency matrices of the graphs.
Our results are applied to discuss the classification problem
for the quantum lens spaces defined by Hong and Szymański,
and to complete the classification of graph $C^*$-algebras associated to
all simple graphs with four vertices or less.
Culler and Vogtmann defined a simplicial space $O(g)$ called
outer space to study the outer automorphism group
of the free group $F_g$. Using representation theoretic methods,
we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g,
SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety
of $F_g,$ reproving a result of Morgan and Shalen. Then we show
that every point $v$ contained in a maximal cell of $O(g)$ defines
a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to
a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there
is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g,
SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a
symplectomorphism on a dense, open subset of $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ with respect to natural symplectic structures
on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this
way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$,
and we show that each $v$ defines a topological decomposition
of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition
of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that
the valuations coming from the closure of a maximal cell in $O(g)$
all arise as divisorial valuations built from an associated projective
compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$
We introduce the tracial Rokhlin property for a conditional expectation
for an inclusion
of unital C*-algebras $P \subset A$ with index finite, and show
that an action $\alpha$
from a finite group $G$ on a simple unital C*-algebra $A$ has
the tracial Rokhlin property
in the sense of N. C. Phillips
if and only if the canonical conditional expectation $E\colon
A \rightarrow A^G$ has the tracial
Rokhlin property.
Let $\mathcal{C}$ be a class of infinite dimensional stably
finite separable unital C*-algebras
which is closed under the following conditions:
(1)
If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$.
(2)
If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$.
(3)
If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection,
then $pAp \in \mathcal{C}$.
Suppose that any C*-algebra in $\mathcal{C}$ is weakly semiprojective.
We prove that if $A$ is a local tracial $\mathcal{C}$-algebra
in the sense of Fan and Fang and a conditional expectation
$E\colon A \rightarrow P$ is of index-finite type with the tracial
Rokhlin property, then $P$ is a unital
local tracial $\mathcal{C}$-algebra.
The main result is that if $A$ is simple, separable, unital
nuclear, Jiang-Su absorbing
and $E\colon A \rightarrow P$ has the tracial Rokhlin property,
then $P$ is Jiang-Su absorbing.
As an application, when an action $\alpha$
from a finite group $G$ on a simple unital C*-algebra $A$ has
the tracial Rokhlin property,
then for any subgroup $H$ of $G$ the fixed point algebra $A^H$
and the crossed product algebra
$A \rtimes_{\alpha_{|H}} H$ is Jiang-Su absorbing.
We also show that the strict comparison property for a Cuntz
semigroup $W(A)$
is hereditary to $W(P)$ if $A$ is simple, separable, exact,
unital,
and $E\colon A \rightarrow P$ has the tracial Rokhlin property.
We consider three point positive masses moving on $S^2$ and $H^2$.
An Eulerian-relative equilibrium, is a relative equilibrium where
the three masses are on the same geodesic, in this paper we analyze
the spectral stability of these kind of orbits where the mass
at the middle is arbitrary and the masses at the ends are equal
and located at the same distance from the central mass. For the
case of $S^2$, we found a positive measure set in the set of
parameters where the relative equilibria are spectrally stable,
and we give a complete classification of the spectral stability
of these solutions, in the sense that, except on an algebraic
curve in the space of parameters, we can determine if the corresponding
relative equilibria is spectrally stable or unstable.
On $H^2$, in the elliptic case, we prove that generically all
Eulerian-relative equilibria are unstable; in the particular
degenerate case when the two equal masses are negligible we get
that the corresponding solutions are spectrally stable. For the
hyperbolic case we consider the system where the mass in the
middle is negligible, in this case the Eulerian-relative equilibria
are unstable.
For a Shimura variety of Hodge type with hyperspecial level
structure at a prime~$p$, Vasiu and Kisin constructed a smooth
integral model (namely the integral canonical model) uniquely
determined by a certain extension property. We define and study
the Ekedahl-Oort stratifications on the special fibers of those
integral canonical models when $p\gt 2$. This generalizes
Ekedahl-Oort stratifications defined and studied by Oort on moduli
spaces of principally polarized abelian varieties and those
defined and studied by Moonen, Wedhorn and Viehmann on good
reductions of Shimura varieties of PEL type. We show that the
Ekedahl-Oort strata are parameterized by certain elements $w$ in
the Weyl group of the reductive group in the Shimura datum. We
prove that the stratum corresponding to $w$ is smooth of dimension
$l(w)$ (i.e. the length of $w$) if it is non-empty. We also
determine the closure of each stratum.