We study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

The present paper is concerned with the study of the ${{L}^{2}}$ cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial ${{L}^{2}}$ cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms.

The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve $L$-function at $s\,=\,1$. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the $L$-function and that the constant in the asymptotics has an unexpected factor of $\sqrt{2}$. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical $L$-function along its critical line. The general $\sqrt{2}$ phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seems much deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

We investigate exceptional sets in the Waring–Goldbach problem. For example, in the cubic case, we show that all but $O\left( {{N}^{79/84+\in }} \right)$ integers subject to the necessary local conditions can be represented as the sum of five cubes of primes. Furthermore, we develop a new device that leads easily to similar estimates for exceptional sets for sums of fourth and higher powers of primes.

Let $E/\mathbb{Q}$ be an elliptic curve defined by the equation ${{y}^{2}}\,=\,{{x}^{3}}\,+\,ax\,+\,b.$ For a prime $p$, $p\nmid \Delta \,=\,-16\left( 4{{a}^{3}}+27{{b}^{2}} \right)\,\ne \,0,$ define

$${{N}_{p}}=p+1-{{a}_{p}}=\,\left| E\left( {{\mathbb{F}}_{p}} \right) \right|.$$

As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant $c$,

$$\prod\limits_{p\le x,p\nmid \Delta }{\frac{{{N}_{p}}}{p}\,\sim \,c{{\left( \log x \right)}^{r}},\,\,\,x\to \infty .}$$

Let ${{\alpha }_{p}}$ and ${{\beta }_{p}}$ be the eigenvalues of the Frobenius at $p$. Define

$${{\tilde{c}}_{n}}=\left\{ \begin{align} & \frac{\alpha _{p}^{k}+\beta _{P}^{k}}{k}\,\,\,\,n={{p}^{k}},\,p\,\text{is}\,\text{a}\,\text{prime,}\,k\,\text{is}\,\text{a}\,\text{natural}\,\text{number},\,p\nmid \Delta . \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \\ \end{align} \right.\,$$

and $\tilde{C}\left( x \right)\,=\,\sum\nolimits_{n\le x}{{{{\tilde{c}}}_{n}}.}$ In this paper, we establish the equivalence between the conjecture and the condition $\tilde{C}\left( x \right)\,=\,\mathbf{o}\left( x \right).$ The asymptotic condition is indeed much deeper than what we know so far or what we can know under the analogue of the Riemann hypothesis. In addition, we provide an oscillation theorem and an $\Omega $ theorem which relate to the constant $c$ in the conjecture.

For a given elliptic curve $\mathbf{E}$, we obtain an upper bound on the discrepancy of sets of multiples ${{z}_{s}}G$ where ${{z}_{s}}$ runs through a sequence $Z\,=\,\left( {{z}_{1}},\ldots ,{{z}_{T}} \right)$ such that $k{{z}_{1}},\ldots ,k{{z}_{T}}$ is a permutation of ${{z}_{1}},\ldots ,{{z}_{T}}$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$.

We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.

Let $A$ be an amenable separable ${{C}^{*}}$-algebra and $B$ be a non-unital but $\sigma $-unital simple ${{C}^{*}}$- algebra with continuous scale. We show that two essential extensions ${{\tau }_{1}}$ and ${{\tau }_{2}}$ of $A$ by $B$ are approximately unitarily equivalent if and only if

$$\left[ {{\tau }_{1}} \right]\,=\,\left[ {{\tau }_{2}} \right]\,\text{in}\,KL\left( A,\,M\left( B \right)/B \right).$$

If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL\left( A,\,M\left( B \right)/B \right)$. Using $KL\left( A,\,M\left( B \right)/B \right)$, we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

In this paper, we find configurations of points in $n$-dimensional projective space $\left( {{\mathbb{P}}^{n}} \right)$ which simultaneously generalize both $k$-configurations and reduced 0-dimensional complete intersections. Recall that $k$-configurations in ${{\mathbb{P}}^{2}}$ are disjoint unions of distinct points on lines and in ${{\mathbb{P}}^{n}}$ are inductively disjoint unions of $k$-configurations on hyperplanes, subject to certain conditions. Furthermore, the Hilbert function of a $k$-configuration is determined from those of the smaller $k$-configurations. We call our generalized constructions ${{k}_{D}}$-configurations, where $D\,=\,\left\{ {{d}_{1}},\ldots ,{{d}_{r}} \right\}$ (a set of $r$ positive integers with repetition allowed) is the type of a given complete intersection in ${{\mathbb{P}}^{n}}$. We show that the Hilbert function of any ${{k}_{D}}$-configuration can be obtained from those of smaller ${{k}_{D}}$-configurations. We then provide applications of this result in two different directions, both of which are motivated by corresponding results about $k$-configurations.

We investigate the problem of whether every immersed flat plane in a nonpositively curved square complex is the limit of periodic flat planes. Using a branched cover, we reduce the problem to the case of $\mathcal{V}\mathcal{H}$-complexes. We solve the problem for malnormal and cyclonormal $\mathcal{V}\mathcal{H}$-complexes. We also solve the problem for complete square complexes using a different approach. We give an application towards deciding whether the elements of fundamental groups of the spaces we study have commuting powers. We note a connection between the flat approximation problem and subgroup separability.