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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
2
votes
Is there a hyperplane avoiding two independent sets?
Not an answer, but this might get you more help by phrasing it in terms of a common combinatorics problem - finding a lower bound for the size of a transversal of a hypergraph. A hypergraph is a colle …
5
votes
Algebraic Geometry for non-mathematician
If you are comfortable doing a lot of exercises, and even better if you can work with a group, then I recommend Ravi Vakil's notes for his course, which are frequently updated and can be found here: h …
9
votes
Stacks in modern number theory/arithmetic geometry
One big recent example would be Lafforgue's proof of the Langlands correspondence for $GL_n$ of function fields (http://arxiv.org/abs/math.NT/0212399), which uses stacks of schtukas. It is similar to …
18
votes
Why tropical geometry?
Grigory Mikhalkin has a few papers which you might find motivational.
To start, a survey on tropical curves from the AMS Notices "What is ..." column from April 2007, available here http://www.ams.or …