Geometrization in Geometry

TL;DR

This survey explains the Geometrization Theorem, a major 21st-century mathematical breakthrough that generalizes the Poincaré Conjecture and has significant implications in differential geometry.

Contribution

It provides an accessible explanation of the Geometrization Theorem and illustrates its applications in differential geometry, highlighting its importance in modern mathematics.

Findings

The Geometrization Theorem unifies various 3-manifold geometries.

It generalizes the Poincaré Conjecture.

Applications include new results in differential geometry.

Abstract

So far, the most magnificent breakthrough in mathematics in the 21st century is the Geometrization Theorem, a bold conjecture by William Thurston (generalizing Poincar\'e's Conjecture) and proved by Grigory Perelman, based on the program suggested by Richard Hamilton. In this survey article, we will explain the statement of this result, also presenting some examples of how it can be used to obtain interesting results in differential geometry.