#
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.

Workshop Videos