We report on theorems by T. Okuyama about complexes generalizing the Coxeter complex and the action of parabolic subgroups on them, both for finite BN-pairs and finite dimensional Hecke algebras. Several simplifications, using mainly the surjections of [CaRi], allow a more compact treatment than the one in [O].

Using vector fields on logarithmic jet spaces we obtain some new positive results for the logarithmic Kobayashi conjecture about the hyperbolicity of complements of curves in the complex projective plane. We are interested here in the cases where logarithmic irregularity is strictly smaller than the dimension. In this setting, we study the case of a very generic curve with two components of degrees $d_{1} \leq d_{2}$ and prove the hyperbolicity of the complement if the degrees satisfy either $d_{1} \geq 4$, or $d_{1} = 3$ and $d_{2} \geq 5$, or $d_{1} = 2$ and $d_{2} \geq 8$, or $d_{1} = 1$ and $d_{2} \geq 11$. We also prove that the complement of a very generic curve of degree $d$ at least equal to 14 in the complex projective plane is hyperbolic, improving slightly, with a new proof, the former bound obtained by El Goul.

Some complex $n$-tori admit holomorphic foliations of codimension one besides the flat ones. It will be shown that such nonlinear foliations, possibly with singularities, can be reduced to those on 2-tori under some topological conditions. A crucial step is an application of the Hodge theory on pseudoconvex manifolds.

For a continuous nonvanishing complex-valued function $g$ on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol $G$ to be semi-Fredholm in terms of mean winding numbers of $\det G$. The matrix function $G$ is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

Let $M \ge 1$ be a positive number. Let $\mathcal{F}$ be a family of holomorphic functions $f$ in some domain $D \subset \mathbb{C}$ for which there exists an integer $k = k(f) \ge 2$ such that $|(f^{k})'(\zeta)| \le M^{k}$ for every periodic point $\zeta$ of period $k$ of $f$ in $D$. We show first that $\mathcal{F}$ is quasinormal of order at most one in $D$. This strengthens a result of W. Bergweiler. Secondly, for the case $M = 1$, we prove that $\mathcal{F}$ is normal in $D$ if there exists a positive number $K < 3$ such that $|f'(\eta)| \le K$ for each $f \in \mathcal{F}$ and every fixed point $\eta$ of $f$ in $D$. This improves a result of M. Essén and S. J. Wu. We also construct an example which shows that the condition $|f'(\eta)| \le K < 3$ can not be replaced by $|f'(\eta)| < 3$.

Let $K$ be an imaginary quadratic field and let $F$ be an abelian extension of $K$. It is known that the order of the class group $\textup{Cl}_{F}$ of $F$ is equal to the order of the quotient $U_{F}/El_{F}$ of the group of global units $U_{F}$ by the group of elliptic units $El_{F}$ of $F$. We introduce a filtration on $U_{F}/El_{F}$ made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

We study a quadruple sequence and express its common limit by Lauricella's hypergeometric function $F_{D}(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, 1; z_{1}, z_{2}, z_{3})$ of three variables. We give a functional equation of $F_{D}$, which is the key to get our expression of the common limit.

We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group $U_{h}\mathfrak{gl}_{n}$. We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

Let $(W, S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W, S)$ and let $\varphi$ be a weight function which is invariant under $G$. Let $\varphi_{G}$ denote the weight function on $W^{G}$ obtained by restriction from $\varphi$. The aim of this paper is to compare the $\mathbf{a}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated with $(W, \varphi)$ and to $(W^{G}, \varphi_{G})$, provided that Lusztig's Conjectures hold.

This paper classifies Enriques surfaces whose K3-cover is a fixed Picard-general Jacobian Kummer surface. There are exactly 31 such surfaces. We describe the free involutions which give these Enriques surfaces explicitly. As a biproduct, we show that $\operatorname{Aut}(X)$ is generated by elements of order 2, which is an improvement of the theorem of S. Kondo.