Applications are closed
for this workshop

## Moduli spaces for algebraic dynamical systems

September 27 to October 1, 2021

at the

American Institute of Mathematics, San Jose, California

organized by

Laura DeMarco, Clay Petsche, and Joseph Silverman

This workshop, sponsored by AIM and the NSF, will be devoted to the study of the geometry and arithmetic of moduli spaces associated to dynamical systems on algebraic varieties, with a particular emphasis on dynamical systems on projective space.

The main topics for the workshop are:

• The geometry of $M(N,d),$ the moduli space of degree d endomorphisms of $P^N.$
• Arithmetic problems associated to rational points on $M(N,d).$
• Dynamical moduli spaces associated to non-rational varieties, such as $K3$ surfaces, that admit non-trivial endomorphisms.
The moduli spaces $M(N,d)$ are dynamical analogues of classical moduli spaces for curves and abelian varieties. They were constructed for $N=1$ in the 1990s, for all $N$ in the 2000s, and with portrait level structure $P$ quite recently, but we have only fragmentary knowledge of their geometry and arithmetic. Among the geometric questions that we hope to address during the workshop are compactifications of $M(N,d),$ the structure of the automorphism locus and the (generalized) critically finite locus, and geometric properties of portrait moduli spaces $M(N,d,P)$ as $P$ increases in complexity. Arithmetic questions that we plan to study include the analysis of the loci in $M(N,d,P)$ associated to uniform boundedness, good reduction, and field of moduli versus field of definition problems.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.