# 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

## Original Announcement

This workshop 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.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.