(CIRM 2022) A translation surface $M$ has an underlying Riemann surface structure $X$. The translation structure identifies to a non-zero Abelian differential on $X$, namely an element $\omega$ of $\Omega(X)$. The space $\Omega(X)$ is a $g$ dimensional vector space that can be canonically identified to $H^1(X; R)$ via $\omega \mapsto [\Re(\omega)]$.
TODO: implement a function that given an element in $H^1(X; R)$ return the corresponding $\eta$ in $\Omega(X)$.
The strategy consists in identifying $\Omega(X)$ to a subspace of meromorphic functions on $X$ via $\eta \mapsto \eta / \omega$. The goal is then to compute $f = \eta / \omega$. To do so, we write symbolically the Taylor expansion of $f$ at a point $z_0 \in X$. The fact that $f \omega$ must be globally defined on $X$ and should integrate according to the given element of $H^1(X; R)$ give enough equation to determine uniquely the Taylor expansion.
(CIRM 2022) A translation surface $M$ has an underlying Riemann surface structure $X$. The translation structure identifies to a non-zero Abelian differential on $X$, namely an element $\omega$ of $\Omega(X)$. The space $\Omega(X)$ is a $g$ dimensional vector space that can be canonically identified to $H^1(X; R)$ via $\omega \mapsto [\Re(\omega)]$.
TODO: implement a function that given an element in $H^1(X; R)$ return the corresponding $\eta$ in $\Omega(X)$.
The strategy consists in identifying $\Omega(X)$ to a subspace of meromorphic functions on $X$ via $\eta \mapsto \eta / \omega$. The goal is then to compute $f = \eta / \omega$. To do so, we write symbolically the Taylor expansion of $f$ at a point $z_0 \in X$. The fact that $f \omega$ must be globally defined on $X$ and should integrate according to the given element of $H^1(X; R)$ give enough equation to determine uniquely the Taylor expansion.