Algebraic Geometry and Topology
How can we tell if two shapes are the same? What if these shapes are not two-dimensional but, say, thirty-three dimensional?
Saying if two shapes are the same is a very difficult question, but it turns out that telling if two shapes are not the same is a slightly easier question!
What do I mean by this? Suppose you have to meet a stranger at the airport. You've never met this person in your life; you don't want to look silly standing there holding a sign. You know the person you are looking for is five foot five, dark haired, wearing red. Suppose you're at the airport and a six foot blonde giant walks out in a green cowboy outfit. You know this is not the person you are looking for. Now suppose, a very neat five foot five brunette walks out wearing a stylish red jumper. Do you know if this is the person you are looking for?
When it comes to wild and wacky shapes it is not easy to attach labels like "red" or "dark haired". This is where algebra comes in. Abstract algebra (like, for example, groups) can be "attached" to some spaces in a natural way; they can function as labels like "red". If two spaces have different algebraic labels, they must be different.
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