most recent 30 from http://mathoverflow.net 2013-05-19T00:18:01Z http://mathoverflow.net/feeds/question/99352 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99352/important-open-questions-in-the-field-of-tropical-geometry john 2012-06-12T09:44:37Z 2012-07-29T07:32:01Z

<p>What are some of the important unanswered questions in the field of tropical geometry?</p>

http://mathoverflow.net/questions/99352/important-open-questions-in-the-field-of-tropical-geometry/103421#103421 Joseph O'Rourke 2012-07-29T02:46:54Z 2012-07-29T02:46:54Z

<p>Three notes:</p> <p>(1) There was an earlier MO question on "<a href="http://mathoverflow.net/questions/84629/learning-tropical-geometry" rel="nofollow">Learning Tropical Geometry</a>."</p> <p>(2) Bernd Sturmfels is giving <a href="http://www.maa.org/mathfest/ia.html#hendrick" rel="nofollow">a series of three lectures</a> on "Algebraic Geometry: Tropical, Convex, and Applied" at the <a href="http://www.maa.org/mathfest/" rel="nofollow">MathFest conference</a> next week.</p> <p>(3) Most directly to your question, see the engaging and succinct 2006 2-page manifesto, "Ten Questions in Tropical Geometry," <a href="http://www.igt.uni-stuttgart.de/AbGeoTop/Feichtner/questions.pdf" rel="nofollow">PDF download</a>:</p> <blockquote> <p>Question 1. Is it possibly true that every tropical variety is shellable?</p> <p>Question 2. Give a good formulation for the moduli space of curves of degree $d$ and genus $g$ lying in $\mathbb{TP}^2$.</p> <p>Question 3. Investigate matroid subdivisions.</p> <p>Question 4. Classify all “root system polytopes”,...</p> <p>Question 5. Compute the (positive) tropical flag variety $GL_4/B$ in its Plücker embedding.</p> <p>Question 6. What do the face lattices of tropical polytopes look like?</p> <p>Question 7. When does tropicalization commute with intersection?</p> <p>Question 8. What is the best axiomatization of tropical oriented matroids?</p> <p>Question 9. Is the tropical discriminant of a defective point configuration a subfan of the secondary fan?</p> <p>Question 10. How can you tell if something is a tropical variety?</p> </blockquote>