#
Arithmetic reflection groups and crystallographic packings

March 7 to March 11, 2022
at the

American Institute of Mathematics,
San Jose, California

organized by

Daniel Allcock,
Alex Kontorovich,
and Alice Mark

## Original Announcement

This workshop will be devoted to the complete classification of maximal hyperbolic arithmetic reflection groups. About 15 years ago, the number of such was shown to be finite, leaving open the possibility of fully determining all such, in a way that makes them accessible for other applications, e.g., the study of crystallographic sphere packings. Recent advances, both theoretical and algorithmic, make it plausible that with sufficient collaborative effort by experts in diverse areas from geometry/topology, dynamics, arithmetic groups, and number theory, will be able to complete this program. In the 2020 online version of this workshop, three groups made progress formulating and resolving questions. In particular, the question of when an Apollonian-type group preserves a quadratic form was answered, and bounds were found on the parameters that can be used to determine the signature of that form.
The main topics of the workshop are:

- Restricting the search space by, on one hand combinatorial and geometric methods, and on the other, spectral gap (Ramanujan) methods

- Computation methods in arithmetic groups

- Vinberg-type and related algorithms

## Material from the workshop

A list of participants.
A report on the workshop activities.