For any family of $N\,\times \,N$ random matrices ${{\left( {{\text{A}}_{k}} \right)}_{k\in K}}$ that is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type $\text{Tr}\left( {{\text{A}}_{k}}\text{M} \right)$, where the matrix $\text{M}$ is deterministic (such random variables include, for example, the normalized matrix entries of ${{\text{A}}_{k}}$). A consequence is the asymptotic independence of the projection of the matrices ${{\text{A}}_{k}}$ 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 $\text{Cu}$ aiming to lift results to the ${{\text{C}}^{\text{*}}}$-algebraic setting. We introduce a new comparison property and relate it to both the corona factorization property $\left( \text{CFP} \right)$ and $\omega $-comparison. We show differences of all properties by providing examples that suggest that the corona factorization for ${{\text{C}}^{\text{*}}}$-algebras might allow for both finite and infinite projections. In addition, we show that Rørdam's simple, nuclear ${{\text{C}}^{\text{*}}}$-algebra with a finite and an inifnite projection does not have the $\text{CFP}$.

We study the Bishop-Phelps-Bollobàs property $\left( \text{BPBp} \right)$ for compact operators. We present some abstract techniques that allow us to carry the $\text{BPBp}$ for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$ and $Y$ be Banach spaces. If $\left( {{c}_{0}},Y \right)$ has the $\text{BPBp}$ for compact operators, then so do $\left( {{C}_{0}}\left( L \right),Y \right)$ for every locally compact Hausdorff topological space $L$ and $\left( X,\,Y \right)$ whenever ${{X}^{*}}$ is isometrically isomorphic to ${{\ell }_{1}}$. If ${{X}^{*}}$ has the Radon-Nikodým property and $\left( {{\ell }_{1}}\left( X \right),\,Y \right)$ has the $\text{BPBp}$ for compact operators, then so does $\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$ for every positive measure $\mu $; as a consequence, $\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$ has the $\text{BPBp}$ for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X={{c}_{0}}$ or $X={{L}_{p}}\left( v \right)$ for any positive measure $v$ and $1\,<\,p\,<\,\infty $. For $1\,\le p\,<\,\infty$, if $\left( X,{{l}_{p}}(Y) \right)$ has the $\text{BPBp}$ for compact operators, then so does $\left( X,{{L}_{p}}\left( \mu ,\,Y \right) \right)$ for every positive measure $\mu $ such that ${{L}_{1}}\left( \mu \right)$ is infinite-dimensional. If $\left( X,\,Y \right)$ has the $\text{BPBp}$ for compact operators, then so do $\left( X,\,{{L}_{\infty }}\left( \mu ,\,\,Y \right) \right)$ for every $\sigma $-finite positive measure $\mu $ and $\left( X,\,C\left( K,\,Y \right) \right)$ 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 $H$ a closed subgroup of $G$. Let $\mu $ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu \right)$.

Let ${{\mathbf{F}}_{q}}[T]$ be the ring of polynomials over the finite field of $q$ elements and $Y$ 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 $\text{exp}\left( {{s}^{1/6-\in }}\,\text{log}\,\text{q} \right)$ solutions.

For a finite abelian $p$-group $A$ of rank $d\,=\,\dim\,A/pA$, let ${{\mathbb{M}}_{A}}\,:=\,\text{lo}{{\text{g}}_{p}}\,{{\left| A \right|}^{1/d}}$ be its (logarithmic) mean exponent. We study the behavior of the mean exponent of $p$-class groups in pro-$p$ towers $\text{L/K}$ 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 $\underline{\mathbb{M}}\left( G \right)$ attached to a finitely generated pro-$p$ group $G$; when $G\,=\,\text{Gal}\left( \text{L/K} \right)$, where $L$ is the Hilbert $p$-class field tower of a number field $K$, $\underline{\mathbb{M}}\left( G \right)$ measures the asymptotic behavior of the mean exponent of $p$-class groups inside $\text{L/K}$. 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 that changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

The $ER\left( 2 \right)$-cohomology of $B\mathbb{Z}/\left( {{2}^{q}} \right)$ and $\mathbb{C}{{\mathbb{P}}^{n}}$ are computed along with the Atiyah–Hirzebruch spectral sequence for $ER{{\left( 2 \right)}^{*}}\left( \mathbb{C}{{\mathbb{P}}^{\infty }} \right)$. This, along with other papers in this series, gives us the $ER\left( 2 \right)$-cohomology of all Eilenberg–MacLane spaces.

We 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 $\mathcal{F}\,\circ \,\left( -\text{log} \right)$ of germs at ${{0}^{+}}$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.