Elementary school level: "Imagine you have a piece of string. You can stretch it, bend it, and twist it in different ways. The Poincaré conjecture is like a puzzle about how you can turn this string into different shapes without breaking it. It's like asking if you can turn a circle into a sphere without making any cuts."
Middle school level: "Okay, let's think about shapes in three dimensions. The Poincaré conjecture is a question about whether any weird, twisted shape in three dimensions can be turned into a simple, smooth shape, like a sphere. Mathematicians wondered if any shape with no holes, like a donut, could be turned into a sphere without tearing or cutting it."
High school level: "The Poincaré conjecture dives into the realm of topology, which is like geometry but more about the properties that don't change when you stretch or squish objects. It asks whether every simply connected, closed 3-dimensional manifold is homeomorphic to the 3-dimensional sphere. In simpler terms, it's asking whether any shape without holes in three dimensions can be continuously deformed into a sphere."
College level: "The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, is a fundamental question in topology and geometry. It conjectures that any closed, simply connected 3-dimensional manifold is homeomorphic to the 3-dimensional sphere. This means that any shape without holes in three dimensions can be transformed into a sphere without cutting or tearing. Proving this conjecture required sophisticated mathematical techniques and was one of the most famous and challenging problems in mathematics for over a century."
Beyond college level: "The Poincaré conjecture, finally proven by Grigori Perelman in 2003, marked a monumental achievement in mathematics and topology. Perelman's proof involved revolutionary insights into geometric analysis and the Ricci flow, a technique from differential geometry. The resolution of this conjecture not only solved a long-standing mathematical puzzle but also had profound implications for our understanding of high-dimensional spaces and the structure of the universe at a fundamental level."