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I am currently taking the equivalent of a first semester graduate course in algebraic topology (roughly covering the first two chapters of Hatcher and possibly more depending on time). Part of the assessment is learning a result above and beyond the scope of the course and delivering a 20 minute presentation of its proof, or something of similar mathematical content. I was wondering if people had any suggestions for a significant result that would be (mostly) accessible to other graduate students toward the end of such a course. I am particularly interested in learning cohomology, so ideally something in that direction. Given that I have a lot of time, I am not afraid to be fairly ambitious and take on something difficult. If it helps, I am fairly interested in algebraic geometry, although I am not limiting myself to that by any means.

Any suggestions would be appreciated.

Thanks

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If you are interested in algebraic geometry, a good topic could be something related to Čech cohomology. Some very basic sheaf theory is involved in the definition, and as you will see this the basic language of modern algebraic geometry.

A good project could be to understand and sketch the proof of De Rham Theorem via Čech cohomology. If you don't want to understand what De Rham cohomology is, also a proof of the equivalence of the Čech cohomology of the sheaf of locally constant functions with the ordinary cohomology can be interesting.

I think you can find everything in Bott and Tu's book "Differential Forms in Algebraic Topology" or in any other book covering this topic. A good motivational introduction to Čech cohomology (Cousin Problem) can be found in Chapter Zero of Griffiths an Harris book "Principles of Algebraic Geometry".

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