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I'm thinking of whether I should self-study Euclidean, elliptic, and hyperbolic geometry but I don't know where these subjects lead to.

Are they prerequisites to algebraic geometry, differential geometry, projective geometry, etc.? How are they related to other subjects?

user816709
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  • You should study them for their own sake. If you are interested in how they relate to other fields, you'll find out when you study those fields. And you'll study geometry with a goal in mind. For all that, I'd highly recommend Stilwell Geometry of Surfaces. There you'll see an elementary presentation of euclidean/elliptic/hyperbolic geometry through the lens of more advanced concepts. – brainjam Feb 12 '21 at 21:05
  • Thank you for the recommendation, I'll check it out. That said, I'd still really appreciate it if you could give a hint of how Euclidean/non-Euclidean geometry relates to other fields. That would help me get a bigger picture of geometry before I plunge into it. – user816709 Feb 13 '21 at 07:10
  • "The pupil asked what he would gain from learning geometry. So Euclid told his slave to get the pupil a coin so he would be gaining from his studies." – Somos Feb 13 '21 at 19:22

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The short answer is that in mathematics there are often profound (and surprising and gratifying) connections between different areas. You might study topic X for its own intrinsic attractions, or you might get into it only because you're fascinated with another topic or problem Y that has connections to field X.

Beyond that, I'd suggest web searches on the connections between topics. For example, the search "hyperbolic geometry and number theory" turned up lots of results, including

From the latter:

In the middle of the 20th century, mathematicians discovered an astonishing link between reciprocity laws and what seemed like an entirely different subject: the “hyperbolic” geometry of patterns such as M.C. Escher’s famous angel-devil tilings of a disk. This link is a core part of the “Langlands program,” a collection of interconnected conjectures and theorems about the relationship between number theory, geometry and analysis. When these conjectures can be proved, they are often enormously powerful: For instance, the proof of Fermat’s Last Theorem boiled down to solving one small (but highly nontrivial) section of the Langlands program.

Mathematicians have gradually become aware that the Langlands program extends far beyond the hyperbolic disk; it can also be studied in higher-dimensional hyperbolic spaces and a variety of other contexts. Now, Scholze has shown how to extend the Langlands program to a wide range of structures in “hyperbolic three-space” — a three-dimensional analogue of the hyperbolic disk — and beyond. By constructing a perfectoid version of hyperbolic three-space, Scholze has discovered an entirely new suite of reciprocity laws.

and, speaking to whether you learn topic X first, or get led to it by topic/problem Y:

“I understood nothing, but it was really fascinating,” he said.

So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”

So by all means start studying geometry. If you don't like it, don't worry:

  • you may never need it
  • the textbook you're reading is not the right one for you
  • the time when you need it for topic/problem Y may be a better time than right now.

On the other hand, learning it now may make you better prepared to tackle topic Y when the time comes.

Some other interesting reads:

  • Series, The Geometry of Markoff Numbers
  • Stilwell, Geometry of Surfaces
brainjam
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