To any $f \in \C[x_{1}, \ldots, x_{n}] \setminus \C$ with $f(0) = 0$ one can associate the motivic zeta function. Another interesting singularity invariant of $f^{-1}\!\{0\}$ is the zeta function on the level of Hodge polynomials, which is actually just a specialization of the motivic one. In this paper we generalize for the Hodge zeta function the result of Veys which provided for $n = 2$ a complete geometric determination of the poles. More precisely we give in arbitrary dimension a complete geometric determination of the poles of order $n-1$ and $n$. We also show how to obtain the same results for the motivic zeta function.

A locally projective amalgam is formed by the stabilizer $G(x)$ of a vertex $x$ and the global stabilizer $G\{x, y\}$ of an edge (containing $x$) in a group $G$, acting faithfully and locally finitely on a connected graph $\Gamma$ of valency $2^{n}-1$ so that (i) the action is $2$-arc-transitive; (ii) the subconstituent $G(x)^{\Gamma(x)}$ is the linear group $SL_{n}(2) \cong L_{n}(2)$ in its natural doubly transitive action and (iii) $[t, G\{x, y\}] \le O_{2}(G(x) \cap G\{x, y\})$ for some $t \in G\{x, y\} \setminus G(x)$. D. Ž. Djokovic and G. L. Miller [DM80], used the classical Tutte's theorem [Tu47], to show that there are seven locally projective amalgams for $n = 2$. Here we use the most difficult and interesting case of Trofimov's theorem [Tr01] to extend the classification to the case $n \ge 3$. We show that besides two infinite series of locally projective amalgams (embedded into the groups $AGL_{n}(2)$ and $O_{2n}^{+}(2)$) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups $M_{22}$, $M_{23}$, $Co_{2}$, $J_{4}$ and $BM$. For each of the exceptional amalgam $n = 3$, $4$ or $5$.

We compare the hyperbolicity with respect to the Lempert function with the other hyperbolicities in the class of pseudoconvex balanced domains.

We are studying some aspects of the action of Galois groups on the torsor of paths connecting two (possibly tangential) points on a projective line minus a finite number of points. We obtain objects which formally behave like classical iterated integrals and polylogarithms. We formulate an analog of Zagier conjecture for these $l$-adic analogs of iterated integrals and polylogarithms.

Let $\Gamma$ be a discrete cocompact subgroup of $\SL_{2}(\C)$. We conjecture that the quotient manifold $X = \SL_{2}(\C)/\Gamma$ contains infinitely many non-isogenous elliptic curves and prove this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where $\Gamma \cap \SL_{2}(\R)$ is cocompact in $\SL_{2}(\R)$.

Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic $2$- and $3$-folds.

We will answer negatively to the question whether the completeness of infinitely sheeted covering surfaces of the extended complex plane have anything to do with their types being parabolic or hyperbolic. This will be accomplished by giving a one parameter family $\{ W[\alpha] : \alpha \in {\cal A} \}$ of complete infinitely sheeted planes $W[\alpha]$ depending on the parameter set ${\cal A}$ of sequences $\alpha = (a_{n})_{n \geq 1}$ of real numbers $0 < a_{n} \leq 1/2$ $(n \geq 1)$ such that $W[\alpha]$ is parabolic for 'small' $\alpha$'s and hyperbolic for 'large' $\alpha$'s.