I am a 10th grader, I am trying to learn smooth manifolds and group theory, our teacher wants us to select a research topic and make a paper about it, (it can be about anything) and I want my paper to be about group theory or topology. What topics would you suggest, I don't want them to be really easy but I don't want super high-level research topics either. What should I do, how can I select a meaningful topic

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    $\begingroup$ This is quite ambitious for a tenth grader, but maybe you could give an explanation of fundamental groups, like what is the fundamental group of a circle, or of a torus, etc. You could explain how fundamental groups can be used to show that various topological manifolds are not homeomorphic. $\endgroup$ – littleO Nov 14 '18 at 11:34
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    $\begingroup$ Those both lead to quite difficult topics rapidly...it's hard (and probably unwise) to give advice without some idea of what you understand already. To combine the topics, you might want to describe the history of Hilbert's $5^{th}$ Problem, say ...but perhaps that already involves too much machinery? I suggest asking somebody who is familiar with your studies up to this point. $\endgroup$ – lulu Nov 14 '18 at 11:38
  • $\begingroup$ Classification of surfaces is a nice topology topic, so is knot invariants like the determinant. In group theory, the structure theorem for finite abelian groups. $\endgroup$ – Gerry Myerson Nov 14 '18 at 12:04
  • $\begingroup$ You should probably explain what level "10th grader" is and what your background consists of. When I think of "10th grade", my thoughts are on things like high school geometry and a second course in high school algebra (factoring trinomials, quadratic equations, simultaneous linear equations in 2 or 3 unknowns, complex numbers, arithmetic and geometric progressions, and maybe you'd get to an advanced topic like logarithms or matrix multiplication). $\endgroup$ – Dave L. Renfro Nov 14 '18 at 12:36
  • $\begingroup$ Curriculum in turkey is pretty light, I have only seen trigonometry and high school geometry, we just finished functions. But I am self-studying since the beginning of 9th grade, I am pretty good at calculus,(Calc2 level) and nowadays I am studying group theory and manifolds from some textbooks and lecture notes I found from internet, I can understand abstract algebra and topology with their proofs without serious problems. (At least beginning level topology and algebra since I am learning) $\endgroup$ – alper akyuz Nov 14 '18 at 13:11

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