We describe the image of the absolute Galois group acting on the pro-$l$ completion of the fundamental group of the $\mathbb {G}_m$ minus $N$th roots of unity. We relate the structure of the image with geometry and topology of modular varieties for the congruence subgroups $\Gamma_1(m;N)$ of ${\rm GL}_m(\mathbb {Z})$ for $m=1,2,3\ldots$.

An injective operator space $V$ which is dual as a Banach space has the form $eR(1-e)$, where $R$ is an injective von Neumann algebra and where $e$ is a projection in $R$. This is used to show that an operator space $V$ is nuclear if and only if it is locally reflexive and $V^{\ast\ast}$ is injective. It is also shown that any exact operator space is locally reflexive.

We give a proof of a result of D. Peterson identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of ${\rm GL}_n$. The totally positive part of this subvariety is then constructed, and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of C. Vafa, K. Intriligator, and A. Bertram for the structure constants (Gromov-Witten invariants). Finally, we use the positivity of these Gromov-Witten invariants to prove certain inequalities for Schur polynomials at roots of unity.

Let $\rho^\prime=\beta^\prime+i\gamma^\prime$ denote the zeros of $\zeta^\prime(s),s=\sigma+it$. It is shown that there is a positive proportion of the zeros of $\zeta^\prime(s)$ in $0<t<T$ satisfying $\beta^\prime-1/2\ll(\log T)^{-1}$. Further results relying on the Riemann hypothesis and conjectures on the gaps between the zeros of $\zeta(s)$ are also obtained.