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.[-]

Given two algebraic ODEs, is there a differential-algebraic relation between generic tuples of their solutions? In recent work with Freitag and Moosa, we produce a bound on the length of tuples one must look at to f ind a relation. Our proof relies on two ingredients. The first is differential Galois theory, combined with the recent proof by Freitag and Moosa of the Borovik-Cherlin conjecture in algebraically closed fields. The second is some general model theory result which allows us to factor any relation through some minimal ODE. I will give a precise statement of our result and sketch the proof. I will also explain why our bound is tight.[-]

The exponential period conjecture predicts how the Galois group of an exponential motive governs all polynomial relations among its periods. For classical motives (which are special exponential motives) this conjecture specialises to the classical period conjecture. My aim is to present some elementary, yet elucidative examples of exponential motives and periods which illustrate how the exponential period conjecture implies certain popular transcendence conjectures, and how its non classical part is related to the Siegel-Shidlovskii theorem.[-]

The density theorem of Schlesinger ensures that the monodromy group of a differential system with regular singular points is Zariski-dense in its differential Galois group. We have analogs of this result for difference systems such as q-difference and Mahler systems, whose only assumption is the regular singular condition. Moreover, solutions of difference or differential systems with regular singularities have good analytical properties. For example, the solutions of differential systems which are regular singular at 0 have moderate growth at 0. We have general algorithms for recognizing regular singularities and they apply to many systems such as differential and q-difference systems. However, they do not apply to the Mahler case, systems that appear in many areas like automata theory. We will explain how to recognize regular singularities of Mahler systems. It is a joint work with Colin Faverjon.[-]

Difference algebraic groups, i.e, groups defined by algebraic difference equations occur naturally as the Galois groups of linear differential or difference equations depending on a discrete parameter. If the linear equation has a full set of algebraic solutions, the corresponding Galois group is an étale difference algebraic group. Like étale algebraic groups can be described as finite groups with a continuous action of the absolute Galois group of the base field, étale difference algebraic groups can be described as certain profinite groups with some extra structure. I will present a decomposition theorem for étale difference algebraic groups, which shows that any étale difference algebraic group can be build from étale algebraic groups and finite groups equipped with an endomorphism.[-]