Let $G$ be the group generated by the transformations $x=\alpha \tilde{x}+b$, $y=\tilde{y}$, $\alpha \ne 0$, $\alpha ,b\in \mathbf{k}$, $\text{char }\mathbf{k}$ of the affine plane ${\mathbf{k}}^2$. For affine algebraic plane curves of the form $y^n=f\left(x\right)$ we reduce a calculation of its $G$-invariants to calculation of the intersection of kernels of some locally nilpotent derivations. We compute a complete set of independent invariants and then reconstruct a curve from given values of these invariants.

The moduli space ${\mathcal{M}}_g$ of compact Riemann surfaces of genus $g$ has orbifold structure, and the set of singular points of such orbifold is the branch locus ${\mathcal{B}}_g$. Given a prime number $p\ge 7$, ${\mathcal{B}}_g$ contains isolated strata consisting of $p$-gonal Riemann surfaces for genera $g\ge \frac{3(p-1)}{2}$, that are multiple of $\frac{p-1}{2}$. This is a generalization of the results obtained in [BCI1] for pentagonal Riemann surfaces, and the results of [K] and [CI3] for zero- and one-dimensional isolated strata in the branch locus.

We study the connectedness of the graph-directed self-affine tiles associated to $\beta$−expansions, called Pisot dual tilings. These tiles are examples of Rauzy fractals and play an important role in the study of $\beta$-expansion, substitution and symbolic dynamical system. Using the complete classification of the $\beta$-expansion of $1$ for quartic Pisot units and the classification of the connected tilings given in [4] and [5], here we continue studying connectedness of Pisot dual tilings generated by a Pisot unit with integral minimal equation $x^4-a{x}^3-b{x}^2-cx-1=0$ in the special case when $a+c-2\lfloor \beta \rfloor$. It is shown that every tile is disconnected having infinitely many connected components.

The Schottky Problem is to characterize the Jacobians in the space of principally polarized abelian varieties (PPAVs). Besides its intrinsic interest, it is related to deep questions in PDE and Physics. The perspective of Information Theory leads to the numerical investigation of properties of the distribution of the periods, especially in cases of relatively large genus. In turn, numerical results lead to conjectures that the periods of hyperelliptic curves are band-limited, and that the squared moduli of the periods of a general surface are Zipfian.