Classifying Manifolds

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 — and even the basic question of 'what kinds of manifolds are possible in each dimension' remains an active area of research. Over the past almost two centuries, incredible progress has been made on this classification. This has resulted in a rather clear picture in low dimensions, and a clear demarcation of the obstacles we must confront as the dimension increases. And while the arguments involved can get quite technical, the big-picture story is a beautiful interplay of shape, symmetry and geometry which deserves to be more widely known.