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What are some of the important unanswered questions in the field of tropical geometry?

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I see some votes to close. (I vote to keep open) I find this question interesting, with the possibility of informative answers. Tropical geometry is a young field of mathematics, and there might be interesting open questions that are not too difficult to solve. That said, I think that this question should be made community wiki. John: could you please do that? – André Henriques Jun 12 '12 at 11:09
Agree with Andre. There are many open problems quests on MO see collection here… – Alexander Chervov Jun 12 '12 at 12:53
There are, but they are of sufficiently narrow scope to be called acceptable for this forum. I think this question does not yet fall in the category of acceptability. If it were made more specific (e.g. "within the last two years", or "that are motivated by field X" where X is sufficiently orthogonal, perhaps combinatorics), then it might be considered acceptable. Gerhard "Ask Me About Being Specific" Paseman, 2012.06.12 – Gerhard Paseman Jun 12 '12 at 15:54
@Gerhard sorry I do not understand you. Other questions are of more broad scope e.g. algebraic geometry or Riemanian geometry. Moreover tropical geometry is young so everything will be "recent years". – Alexander Chervov Jun 12 '12 at 16:16
Perhaps the best way to demonstrate why I voted for this question is to reveal my own ignorance and confusion about this subject. Many younger researchers may carry the same uncertainties. How much of the ``package'' of algebraic geometry carries over to the tropical geometry? When working with the tropical semigroup ring, what topological spaces can you construct from tropical varieties and how exactly do they compare to varieties over C? What general results have been proven in tropical Gromov-Witten theory? If these questions are not open, there must be related ones that are, right? – Eric Zaslow Jun 13 '12 at 14:27

Three notes:

(1) There was an earlier MO question on "Learning Tropical Geometry."

(2) Bernd Sturmfels is giving a series of three lectures on "Algebraic Geometry: Tropical, Convex, and Applied" at the MathFest conference next week.

(3) Most directly to your question, see the engaging and succinct 2006 2-page manifesto, "Ten Questions in Tropical Geometry," PDF download:

Question 1. Is it possibly true that every tropical variety is shellable?

Question 2. Give a good formulation for the moduli space of curves of degree $d$ and genus $g$ lying in $\mathbb{TP}^2$.

Question 3. Investigate matroid subdivisions.

Question 4. Classify all “root system polytopes”,...

Question 5. Compute the (positive) tropical flag variety $GL_4/B$ in its Plücker embedding.

Question 6. What do the face lattices of tropical polytopes look like?

Question 7. When does tropicalization commute with intersection?

Question 8. What is the best axiomatization of tropical oriented matroids?

Question 9. Is the tropical discriminant of a defective point configuration a subfan of the secondary fan?

Question 10. How can you tell if something is a tropical variety?

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