For a polynomial $f(x)$ over a field $L$, and an element $c \in L$, I will discuss the size of the intersection of the orbit $\lbrace f(c),f(f (c)),...\rbrace$ with a prescribed subfield of $L$. I will also discuss the size of the intersection of orbits of two distinct polynomials, and generalizations of these questions to more general maps between varieties.
polynomial decomposition - classification of finite simple groups - Bombieri-Lang conjecture - orbit - dynamical system - unlikely intersections[-]

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.[-]

I will introduce the Speiser and Eremenko-Lyubich classes of transcendental entire functions and give a brief review of quasiconformal maps and the measurable Riemann mapping theorem. I will then discuss tracts and models for the Eremenko-Lyubich class and state the theorem that all topological tracts can occur in this class. A more limited result for the Speiser class will also be given. I will then discuss some applications of these ideas, focusing on recent work with Kirill Lazebnik (prescribing postsingular orbits of meromorphic functions) and Lasse Rempe (equilateral triangulations of Riemann surfaces).[-]

Fatou noticed in 1926 that certain transcendental entire functions have Julia sets in which there are curves of points that escape to infinity under iteration and he wondered whether this might hold for a more general class of functions. In 1989, Eremenko carried out an investigation of the escaping set of a transcendental entire function f, $I(f)=\left \{ z\in\mathbb{C}:\left | f^{n}\left ( z \right ) \right | \rightarrow \infty \right \}$ and produced a conjecture with a weak and a strong form. The strong form asks if every point in the escaping set of an arbitrary transcendental entire function can be joined to infinity by a curve in the escaping set.
This was answered in the negative by the 2011 paper of Rottenfusser, Rückert, Rempe, and Schleicher (RRRS) by constructing a tract that produces a function that cannot contain such a curve. In the same paper, it was also shown that if the function was of finite order, that is, log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )$ as $\left | z \right |\rightarrow \infty$, then every point in the escaping set can indeed be connected to infinity by a curve in the escaping set.
The counterexample $f$ used in the RRRS paper has growth such that log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )^{k}$ where $K > 12$ is an arbitrary constant. The question is, can this exponent, K, be decreased and can explicit calculations and counterexamples be performed and constructed that improve on this?[-]

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.[-]