Abstract

Manifolds are ubiquitous in modern mathematics, from the familiar low dimensional examples of curves and surfaces in calculus, to higher dimensional abstract examples in geometry, physics, data science and beyond.
They come in a bewildering variety - with the basic question of “what kinds of manifolds are possible” providing a powerful guiding light in topology.

While the 19th century witnessed a near complete understanding of the 1 and 2 dimensional cases, much p rogress during the 20th century in 3 dimensions was guided by a conjecture of Poincare, first formulated in 1904. Poincare’s conjecture - essentially that simple 3 dimensional spaces can be probed effectively using 1-dimensional loops - proved much more difficult than originally hoped, remaining unsolved for nearly 100 years.

Following a century of work, its eventual resolution by Perelman in 2002 provided a new and powerful tool - called Geometrization - to the study of all 3-dimensional spaces. And while the arguments involved get quite technical, the big-picture story is a beautiful interplay of shape, symmetry and geometry which deserves to be more widely known. My goal in this talk is to give an overview of this exciting story lying at the heart of modern topology, from what was asked to what the mathematical community has learned.

Occurances

While I have been planning this talk in my head for quite some time, I have so far only had the opportunity to give it once - at ICERM’s summer REU of 2022. Hoping to be able to share the story with more soon!

Slides

Here are the slides from this first version of the talk!