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 define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case, and we define a doubled Rasmussen invariant. This invariant is used to obtain various cobordism obstructions; in particular, it is an obstruction to sliceness. Finally, we show that the doubled Rasmussen invariant contains the odd writhe of a virtual knot and use this to show that knots with non-zero odd writhe are not slice.

In this paper we prove that decomposable forms, or homogeneous polynomials $F\left( {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right)$ with integer coefficients that split completely into linear factors over $\mathbb{C}$, take on infinitely many square-free values subject to simple necessary conditions, and they have $\text{deg}\,f\,\le \,2n\,+\mid 2$ for all irreducible factors $f$ of $F$. This work generalizes a theorem of Greaves.

We prove that for every surface $\Sigma $ of Euler genus $g$, every edge-maximal embedding of a graph in $\Sigma $ is at most $O(g)$ edges short of a triangulation of $\Sigma $. This provides the first answer to an open problem of Kainen (1974).

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.

Let $c$ be a curve of genus at least 2 embedded in ${{E}_{1}}\,\times \,.\,.\,.\,\times \,{{E}_{N}}$ , where the ${{\text{E}}_{i}}$ are elliptic curves for $i\,=\,1,\,.\,.\,.\,,\,N$. In this article we give an explicit sharp bound for the Néron–Tate height of the points of $c$ contained in the union of all algebraic subgroups of dimension $<\,\max ({{r}_{C}}\,-{{t}_{C}},\,{{t}_{C}})$ , where ${{t}_{C}}(\text{resp}\text{.}{{r}_{C}})$ is the minimal dimension of a translate (resp. of a torsion variety) containing $c$.

As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin–Dem’janenko method for curves in products of elliptic curves.

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.

The ${{c}_{2}}$ invariant is an arithmetic graph invariant defined by Schnetz. It is useful for understanding Feynman periods. Brown and Schnetz conjectured that the ${{c}_{2}}$ invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the ${{c}_{2}}$ invariant in the case where we are over the field with 2 elements and the completed graph has an odd number of vertices. The methods involve enumerating certain edge bipartitions of graphs; two different constructions are needed.

In this paper we study a class of second order fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds and obtain a priori estimates under proper assumptions close to optimal. The analysis developed here should be useful to deal with other Hessian equations containing gradient terms in other contexts.

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.