We prove the existence of extremal, nonconstant–scalar curvature, Kähler metrics on certain unstable projectivized vector bundles $\mathbb{P}\left(E\right)\to M$ over a compact constant–scalar curvature Kähler manifold $M$ with discrete holomorphic automorphism group, in certain adiabatic Kähler classes. In particular, the vector bundles $E\to M$ are assumed to split as a direct sum of stable subbundles $E=E_1\oplus \cdots \oplus E_s$ all having different Mumford–Takemoto slope, for example, $\mu \left(E_1\right)>\cdots >\mu \left(E_s\right)$.

We present an algorithm, based on the explicit formula for $L$-functions and conditional on the generalized Riemann hypothesis, for proving that a given integer is squarefree with little or no knowledge of its factorization. We analyze the algorithm both theoretically and practically and use it to prove that several RSA challenge numbers are not squarefull.

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals $(x,x+x^{ϵ}]$ is about $x^{ϵ}/logx$. The second says that the number of primes $p<x$ in the arithmetic progression $p\equiv a\left(\mathrm{mod}d\right)$, for $d<x^{1-\delta }$, is about $\frac{\pi \left(x\right)}{\varphi \left(d\right)}$, where $\varphi$ is the Euler totient function.

More precisely, for short intervals we prove: Let $k$ be a fixed integer. Then

$${\pi }_q\left(I\right(f,ϵ\left)\right)\sim \frac{\#I(f,ϵ)}{k},q\to \infty$$ holds uniformly for all prime powers $q$, degree $k$ monic polynomials $f\in {\mathbb{F}}_q\left[t\right]$ and ${ϵ}_0(f,q)\le ϵ$, where ${ϵ}_0$ is either $\frac{1}{k}$, or $\frac{2}{k}$ if $p\mid k(k-1)$, or $\frac{3}{k}$ if further $p=2$ and $\mathrm{deg}f\text{'}\le 1$. Here $I(f,ϵ)=\{g\in {\mathbb{F}}_q[t]\mid \mathrm{deg}(f-g)\le ϵ\mathrm{deg}f\}$, and ${\pi }_q\left(I\right(f,ϵ\left)\right)$ denotes the number of prime polynomials in $I(f,ϵ)$. We show that this estimation fails in the neglected cases.

For arithmetic progressions we prove: let $k$ be a fixed integer. Then

$${\pi }_q(k;D,f)\sim \frac{{\pi }_q\left(k\right)}{\varphi \left(D\right)},q\to \infty ,$$ holds uniformly for all relatively prime polynomials $D,f\in {\mathbb{F}}_q\left[t\right]$ satisfying $\Vert D\Vert \le q^{k(1-{\delta }_0)}$, where ${\delta }_0$ is either $\frac{3}{k}$ or $\frac{4}{k}$ if $p=2$ and $(f/D)\text{'}$ is a constant. Here ${\pi }_q\left(k\right)$ is the number of degree $k$ prime polynomials and ${\pi }_q(k;D,f)$ is the number of such polynomials in the arithmetic progression $P\equiv f\left(\mathrm{mod}d\right)$.

We also generalize these results to arbitrary factorization types.

We give a new local proof of the Breuil–Mézard conjecture for $2$-dimensional representations of the absolute Galois group of ${\mathbb{Q}}_p$, when $p\ge 5$ and the representation has scalar endomorphisms.

This work builds on the foundation laid by Gordon and Wilson in their study of isometry groups of solvmanifolds—Riemannian manifolds that admit a transitive solvable group of isometries. We restrict ourselves to a natural class of solvable Lie groups called almost completely solvable; this class includes the completely solvable Lie groups. When the commutator subalgebra contains the center, we have a complete description of the isometry group of any left-invariant metric using only metric Lie algebra information.

Using our work on the isometry group of such spaces, we study quotients of solvmanifolds. Our first application is to the classification of homogeneous Ricci soliton metrics. We show that the verification of the generalized Alekseevsky conjecture reduces to the simply connected case. Our second application is a generalization of a result of Heintze on the rigidity of existence of compact quotients for certain homogeneous spaces. Heintze’s result applies to spaces with negative curvature. We remove all the geometric requirements, replacing them with algebraic requirements on the homogeneous structure.