#
Definability and decidability problems in number theory

May 6 to May 10, 2019
at the

American Institute of Mathematics,
San Jose, California

organized by

Kirsten Eisentraeger,
Valentina Harizanov,
Jennifer Park,
and Alexandra Shlapentokh

## Original Announcement

This workshop will be devoted to definability and decidability problems in number theory.
The main topics for the workshop are

- H10 and existential definability of $\mathbb{Z}$ for $\mathbb{Q}$ and big subrings of $\mathbb{Q}$.
- Decidability of the first-order and the existential theory in finite and infinite algebraic extensions of $\mathbb{Q}$.
The workshop will bring together mathematicians working in algebraic geometry, number theory, model theory and computability theory to work on problems in decidability/computability and definability in number theory.
- Definability of valuation rings over infinite algebraic extensions of $\mathbb{Q}$ and over function fields.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

A topological approach to undefinability in algebraic extensions of ${\mathbb Q}$

by Kirsten Eisentraeger, Russell Miller, Caleb Springer, Linda Westrick

Hilbert's Tenth Problem and the Inverse Galois Problem

by Francesca Balestrieri, Jennifer Park, Alexandra Shlapentokh

The Diophantine problem for rings of exponential polynomials

by D. Chompitaki, N. Garcia-Fritz, H. Pasten, T. Pheidas, X. Vidaux

Unique equilibrium states for geodesic flows over surfaces without focal points

by Dong Chen, Lien-Yung Kao, Kiho Park

Big fields that are not large

by Barry Mazur, Karl Rubin