A personal take on model theory

A central theme of modern mathematical logic is complexity or its absence in various contexts. In particular, model theory often considers a mathematical structure (e.g., a field, a group, or a graph) where we make one or more absence-of-complexity assumptions: its definable sets have simple descriptions, its theory has few models, its definable families have bounded VC-dimension, etc.

Remarkably, such assumptions sometimes lead to the structure exhibiting certain geometric behaviors (e.g., it has a good notion of dimension, its definable groups resemble algebraic groups or Lie groups at different levels). Hence, from another angle, model theory can be seen as developing certain aspects of geometry more generally. As a consequence, the subject provides new tools for solving problems beyond the reach of classical geometries (e.g., algebraic geometry, differential geometry).