RESEARCH PROBLEMS IN TROPICAL GEOMETRY
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Problems from AIM, Palo Alto, October 2003
http://www.aimath.org/WWN/amoebas/

Problems from Atlanta, January 2005
http://math.berkeley.edu/~develin/tropicalproblems.html



Ten Problems suggested by Bernd on August 18, 2005:
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1. Study mixtures of trees and tropical Pfaffian varieties.
   Solve the problem stated at the end of Chapter 3 in [ASCB]

2. Classify all tropical curves of degree at most four in TP^3.

3. Compute the (positive) tropicalization of the Grassmannian G(3,7)
   and all of its Schubert subvarieties and its  positive matroid strata.

4. Characterize reduced Gr"obner bases whose local tropical variety is a sphere.

5. Devise and implement an algorithm for computing the positive tropical variety
   of an ideal. Show that the positive tropical variety given by the 3x3-minors
   of a Hankel matrix is always the union of two circles.

6. Is the positive tropical variety of any Grassmannian a sphere ?
   Same question for flag varieties and their Schubert subvarieties.

7. Identify a minimal tropical basis for any matroid. Do the induced cycles 
   form a tropical basis for (the Bergman fan of) any graphic matroid ?

8. Does every determinantal variety admit a parametrization whose
   tropicalization is the tropicalized determinantal variety  ?

9. Formulate and prove the "tropical Steinitz theorem" for tropical 3-polytopes.
   What about face posets of four-dimensional tropical polytopes ?

10. Does every Minkowski summand of the associahedron appear as a face of some 
    tropical linear space (Problem 2.7 in Speyer's paper math.CO/0410455 ) ?