Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$, where ${P_f}$ is the postcritical set of f. Here the isotopy class $[f^{-1}(\alpha )]$ (relative to ${P_f}$) only depends on the isotopy class $[\alpha ]$. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation. We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

We study cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^{1}$ as intersection numbers of tagged curves and Auslander–Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.