what Manifolds are and what they’re used to understand / interpret / model?
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They’re spaces that can be locally approximated by normal Euclidean spaces (e.g. lines or planes) but can be non-Euclidean on a large scale. A very simple example is the surface of a sphere, which is non-Euclidean because, for example, if you try and draw a triangle, the angles do not sum to 180 degrees. However, we’re all familiar with how a 2D rectangle can very closely approximate the geometry of a small portion of the earth’s surface.
They’re used in a few areas of maths and physics, but perhaps most prominently in general relativity, which models space and time as something called a Lorentzian manifold (obviously the simpler examples, like the surface of a sphere, are used everywhere, but they usually aren’t discussed using the language of manifolds).
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