The cohomology algebra mod $p$ of the complex projective Stiefel manifolds is determined for all primes $p$ . When $p=2$ we also determine the action of the Steenrod algebra and apply this to the problem of existence of trivial subbundles of multiples of the canonical line bundle over a lens space with 2-torsion, obtaining optimal results in many cases.

Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in ${{\text{C}}^{\text{2}}}$ is studied in details.

Let $E$ be an elliptic curve over $\mathbb{Q}$ , and let $r$ be an integer. According to the Lang-Trotter conjecture, the number of primes $p$ such that ${{a}_{p}}\left( E \right)=r$ is either finite, or is asymptotic to ${{C}_{E,r}}\sqrt{x}/\log x$ where ${{C}_{E,r}}$ is a non-zero constant. A typical example of the former is the case of rational $\ell $ -torsion, where ${{a}_{p}}\left( E \right)=r$ is impossible if $r\equiv 1\,\left( \bmod \,\ell\right)$ . We prove in this paper that, when $E$ has a rational $\ell $ -isogeny and $\ell \ne 11$ , the number of primes $p$ such that ${{a}_{p}}\left( E \right)\equiv r\,\left( \bmod \,\ell\right)$ is finite (for some $r$ modulo $\ell $ ) if and only if $E$ has rational $\ell $ -torsion over the cyclotomic field $\mathbb{Q}\left( {{\zeta }_{\ell }} \right)$ . The case $\ell =11$ is special, and is also treated in the paper. We also classify all those occurences.

Let $\Gamma $ be a rank-one arithmetic subgroup of a semisimple Lie group $G$ . For fixed $K$ -Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of $G$ , whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma $ -automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Gamma $ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma $ follows from results of DeGeorge-Wallach and Delorme.

Following the investigations of $\text{B}$ . Abrahamse $[1]$ , F. Forelli $[11]$ , M. Heins $[14]$ and others, we continue the study of the Pick-Nevanlinna interpolation problem in multiply-connected planar domains. One major focus is on the problem of characterizing the extreme points of the convex set of interpolants of a fixed data set. Several other related problems are discussed.

Many aspects of the behavior of averages in ergodic theory are a matter of counting the number of times a particular event occurs. This theme is pursued in this article where we consider large deviations, square functions, jump inequalities and related topics.

In this paper we study properties of the functions which satisfy modular equations for infinitely many primes. The two main results are:

1. 1) every such function is analytic in the upper half plane;

2. 2) if such function takes the same value in two different points ${{z}_{1}}$ and ${{z}_{2}}$ then there exists an $f$ -preserving analytic bijection between neighbourhoods of ${{z}_{1}}$ and ${{z}_{2}}$ .

Let $\tilde{M}$ be a regular branched cover of a homology 3-sphere $M$ with deck group $G\cong \mathbb{Z}_{2}^{d}$ and branch set a trivalent graph $\Gamma $ ; such a cover is determined by a coloring of the edges of $\Gamma $ with elements of $G$ . For each index-2 subgroup $H$ of $G,\,{{M}_{H}}=\tilde{M}/H$ is a double branched cover of $M$ . Sakuma has proved that ${{H}_{1}}\left( {\tilde{M}} \right)$ is isomorphic, modulo 2-torsion, to ${{\oplus }_{H}}{{H}_{1}}\left( {{M}_{H}} \right)$ , and has shown that ${{H}_{1}}\left( {\tilde{M}} \right)$ is determined up to isomorphism by ${{\oplus }_{H}}{{H}_{1}}\left( {{M}_{H}} \right)$ in certain cases; specifically, when $d=2$ and the coloring is such that the branch set of each cover ${{M}_{H}}\to M$ is connected, and when $d=3$ and $\Gamma $ is the complete graph ${{K}_{4}}$ . We prove this for a larger class of coverings: when $d=2$ , for any coloring of a connected graph; when $d=3\,\text{or}\,\text{4}$ , for an infinite class of colored graphs; and when $d=5$ , for a single coloring of the Petersen graph.

The Sierpiński carpets first considered by C.McMullen and later studied by Y. Peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. This paper (i) calculates the Hausdorff, box (or Minkowski), and packing dimensions of the modified Sierpiński carpets and (ii) shows that for these sets the Hausdorff and packing measures in their dimension are never zero and gives necessary and sufficient conditions for these measures to be infinite.

The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.