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May 16th, 2017, 09:56 AM   #1
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How do I become a geometry researcher?

Hi all,

I'm a first year undergrad student. I'm interested in geometry for a number of reasons, but unfortunately my university offers barely any more than the typical introductory modules. As such, I'm left without knowledge of what path to take ; what order to learn things in. My personal tutor is an algebraist and so I already have a good idea of the order in which things are learned on the path to arriving at the "cutting edge". However, I'd like to have an idea of a path for geometry seeing as I'll have to indulge this interest myself. Where should I start? what order to tackle things? what should I even tackle?

Regards,

M
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May 16th, 2017, 09:56 AM   #2
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I should say, I'm planning on reading some of Euclid's Elements this summer. From there, I really have no idea how this all evolves.
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May 16th, 2017, 10:44 AM   #3
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Since modern geometry is mostly group theory and category theory, being an algebraist is the right background.

Since you're an early undergrad, make sure you get a good grounding in calculus and especially multivariable calculus, which is the prereq for differential geometry.
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May 16th, 2017, 10:58 AM   #4
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Quote:
I should say, I'm planning on reading some of Euclid's Elements this summer. From there, I really have no idea how this all evolves.
You would be better employed spending the summer with this undergraduate book

Elementary Geometry

John Roe

Oxford University Press

You would be suprised what Geometry encompasses these days

What aspects are you particularly interested in?
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May 16th, 2017, 03:23 PM   #5
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to studiot,

there are lots of books in the university library that look very interesting. Differential geometry looks to me like a good time, and I gather it's the starting point for a lot of things. I'm doing some topology on the degree next year and looking forward to that. I guess the non euclidean geometry stuff prompted my interest; i thought the ancient axioms weren't debatable and yet apparently things like hyperbolic geometry take a completely different view to things, which naturally gives me questions I want to find the answers to. I have to mention knot theory, purely because it just looks alien and strange. Not sure if that's connected to this all but I sure hope it is. Anything to justify diving into that. Other than that, it's just terms I've heard in passing ... Riemann surfaces? I'm past the sphere of my knowledge by now. I'm currently self teaching some measure theory and Lebesgue integration. I have to say I'm enjoying all that, and would hope for a connection between analysis and geometry. Your comment about geometry encompassing more than expected pretty much sums up this mysterious veil I want to lift off of geometry.

to Mashke,

Your comment surprises me in the sense that I would not have expected that connection. I guess that because I'm only a first year undergrad, and geometry and group theory seemed completely different. I am currently re-teaching myself multi variable calculus proper (the lecturer did not cover proofs). I enjoyed linear algebra and am interested in at least studying some multi linear algebra (tensors etc).

I refuse to reject random areas of pure maths in favour of others, I'm interested in geometry for sure but I think that at the end of the day, the more you poke a subject with a stick, the more likely it will be to jump up and do something crazy that you didn't expect. I'm still in awe of the fundamental theorem of calculus. If I discovered that, I'd be bouncing off of the walls.
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May 16th, 2017, 03:56 PM   #6
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Quote:
Originally Posted by Magnitude View Post
Your comment surprises me in the sense that I would not have expected that connection.
The connection is generally through groups of transformations. Here's a page I found that's pretty good. The big name is Felix Klein. He showed that you can classify geometric spaces by the transformation groups that preserve various properties.

Here's a pretty good page. https://en.wikipedia.org/wiki/Transformation_geometry

I only mentioned this connection is because someone interested in geometry might not pay attention in abstract algebra class because at the elementary level they don't appear to have any relation at all. Group theory is generally taught in a very a-historical manner that hides why people are interested in groups in the first place. One big application of group theory is in geometry.

ps Also see https://en.wikipedia.org/wiki/Erlangen_program

Last edited by Maschke; May 16th, 2017 at 04:07 PM.
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