Quadratic polynomials have been investigated since the beginnings of complex dynamics, and are often approached through combinatorial theories such as laminations or Hubbard trees. I will explain how both of these approaches fit in a more algebraic framework: that of iterated monodromy groups. The invariant associated with a quadratic polynomial is a group acting on the infinite binary tree, these groups are interesting in their own right, and provide insight and structure to complex dynamics: I will explain in particular how the conversion between Hubbard trees and external angles amounts to a change of basis, how the limbs and wakes may be defined in the language of group theory, and present a model of the Mandelbrot set consisting of groups. This is joint work with Dzmitry Dudko and Volodymyr Nekrashevych.[-]

W. Thurston's theorems almost all aim to give a purely topological problem an appropriate geometry, or to identify an appropriate obstruction.. We will illustrate this in two examples:
--The Thurston pullback map to make a rational map from a post-critically finite branched cover of the sphere, and
--The skinning lemma, to find a hyperbolic structure for a Haken 3-manifold.
In both cases, either the relevant map on Teichmüller space has a fixed point, solving the geometrization problem, or there is an obstruction consisting a multicurve.[-]

Thurston's topological characterization theory asks whether there is a holomorphic dynamical system that realizes topological (even combinatorial) data, this often allows to describe all possible dynamical systems in a certain parameter space. I work on extending Thurston topological characterization theory to different classes of transcendental functions. In my talk I will start with some explicit families of functions for which we have established an extension of Thurston's theory, and then describe further extensions by compositions of such functions. The presented ideas are part of my PhD thesis under the supervision of Dierk Schleicher. This is work in progress.[-]

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree 2 circle maps $F_{\lambda}: \mathbb{R} / \mathbb{Z} \rightarrow \mathbb{R} / \mathbb{Z}$ defined by
$$
F_{\lambda}(x):=2 x+a+\frac{b}{\pi} \sin (2 \pi x), \text { with } \lambda:=(a, b) \in \mathbb{R} / \mathbb{Z} \times(0,1)
$$
We prove that if $F_{\lambda o}^{\circ n}-$ id has a zero of multiplicity 3 in $\mathbb{R} / \mathbb{Z}$, then there is a system of local coordinates $(\alpha, \beta): W \rightarrow \mathbb{R}^{2}$ defined in a neighborhood $W$ of $\lambda_{0}$, such that $\alpha\left(\lambda_{0}\right)=\beta\left(\lambda_{0}\right)=0$ and $F_{\lambda}^{\circ n}-$ id has a multiple zero with $\lambda \in W$ if and only if $\beta^{3}(\lambda)=\alpha^{2}(\lambda)$. This shows that the tips of tongues are regular cusps. This is joint work with K. Banerjee, J. Canela and A. Epstein.[-]