The degree of a dominant rational map $f: \mathbb{P}^n \rightarrow \mathbb{P}^n$ is the common degree of its homogeneous components. By considering iterates of $f$, one can form a sequence $\operatorname{deg}\left(f^n\right)$, which is submultiplicative and hence has the property that there is some $\lambda \geq 1$ such that $\left(\operatorname{deg}\left(f^n\right)\right)^{1 / n} \rightarrow \lambda$. The quantity $\lambda$ is called the first dynamical degree of $f$. We'll give an overview of the significance of the dynamical degree in complex dynamics and describe an example of a birational self-map of $\mathbb{P}^3$ in which this dynamical degree is provably transcendental. This is joint work with Jeffrey Diller, Mattias Jonsson, and Holly Krieger.[-]

The automorphism group of a compact Kähler manifold satisfies Tits alternative: any subgroup either admits a solvable subgroup of finite index or contains a free non-abelian group of two generators (Campana-WangZhang). In the first case, this group cannot be too big. Some algebraic (rational) manifolds with special automorphisms admit infinitely many nonequivalent real forms. This talk is based on my (old and recent) works with F. Hu, H.-Y. Lin, V.-A. Nguyen, K. Oguiso, N. Sibony, X. Yu, D.-Q. Zhang.[-]