# Program

This program is focused on the two-way interaction of logical ideas and techniques, such as definability from model theory and decidability from computability theory, with fundamental problems in number theory. These include analogues of Hilbert's tenth problem, isolating properties of fields of algebraic numbers which relate to undecidability, decision problems around linear recurrence and algebraic differential equations, the relation of transcendence results and conjectures to decidability and decision problems, and some problems in anabelian geometry and field arithmetic. We are interested in this specific interface across a range of problems and so intend to build a semester which is both more topically focused and more mathematically broad than a typical MSRI program.

**Keywords and Mathematics Subject Classification (MSC)**

**Tags/Keywords**

number theory

model theory

computability theory

first-order and Diophantine definability

Hilbert's Tenth Problem

Diophantine equations

Diophantine stability

Diophantine geometry

ranks of abelian varieties

field arithmetic

function fields

number fields

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**