MCS 595 Graduate Seminar, Fall 2007

This semester, we meet weekly in 712 SEO on Thursdays at 11AM. The topic of this seminar is Tropical Geometry.
  1. Thu 30 Aug: we made the connections between Newton polytopes and amoebas, observing that the tentacles of the image of a hypersurface under the logarithmic map coincide with the directions of the normals to the faces of the Newton polytope. We ended by sketching the graph of a tropical line.
  2. Thu 6 Sep: regular triangulations of Newton polytopes lead to tropicalizations of hypersurfaces supported on these Newton polytopes. We sketched the correspondence between the outer normal cones to the simplices in a regular triangulation and the tropicalizations. Inner edges reflect the spine of an amoeba.
  3. Thu 13 Sep: we determined the "degree" of a 4-nomial curve by intersecting its tropicalizations with a tropical line. Looking at sums of polytopes, we defined a recipe to assign multiplicities to the intersection points of a tropicalized hypersurface and a tropical line. Generically, the number of such intersection points counted with multiplicity equals the degree of the original hypersurface.
  4. Thu 20 Sep: we tropicalized Bernshtein's theorem justifying it by means of polyhedral homotopies.
  5. Thu 27 Sep: Dimitris Diochnos presented his master's thesis.
  6. Thu 5 Oct: Kathy Piret and Yun Guan gave a preview of their talks at the AMS meeting at DePaul University.
  7. Thu 11 Oct: We examined the special case of two intersecting planar curves and developed algorithms to detect (1) whether their Newton polygons are in general position; and (2) once we know the Newton polygon share a common normal, whether then the polynomials have a common root at infinity.
  8. Thu 18 Oct: As a continuation of last's weeks seminar, we examined how to apply the two certicates of degeneracies, using the same strategy. In case there is a common factor, first compute its support (or certify that it cannot exist) before its coefficients. We ended by considering the polygonal decomposition problem.
  9. Thu 25 Oct: We do not meet at 11AM, but meet at 4PM for the Algebraic Geometry Seminar of Frank Sottile.
  10. Thu 1 Nov: talk of Olga Kashcheyeva.
  11. Thu 8 Nov: We did not meet; attended the talks of Joe Harris.
  12. Thu 15 Nov: presentation of Danko Adrovic on tropical factorization.
  13. Thu 22 Nov: thanksgiving.
  14. Thu 29 Nov: Kathy Piret presents her talk for MACIS'07: computing critical points by continuation.

Seminars organized in previous semesters: