Zilber's notion of logically perfect structure: Universal Covers

TL;DR

This paper explores the intersection of model theory with classical studies of universal covers in complex analysis, algebraic topology, and number theory, highlighting recent advances and unified approaches across various mathematical contexts.

Contribution

It provides a systematic and unified framework for understanding universal covers in different mathematical settings using model theory.

Findings

Unified approach to exponential, modular, and Shimura covers

Insights into canonicity of universal covers across contexts

Methodological advancements in analyzing complex covers

Abstract

We sketch recent interactions between model theory and a roughly 150-year old study of analytic functions involving complex analysis, algebraic topology, and number theory, centered in canonicity of universal covers. Towards this goal we discuss in a systematic and unified way several examples indicating the main ideas of the proofs and the necessary changes in method for different situations: exponential covers, modular and Shimura curves, Shimura and abelian varieties, and coherent families of smooth covers.