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.
- 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.
- 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.
- 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.
- Thu 20 Sep: we tropicalized Bernshtein's theorem
justifying it by means of polyhedral homotopies.
- Thu 27 Sep: Dimitris Diochnos presented his master's thesis.
- Thu 5 Oct: Kathy Piret and Yun Guan gave a preview of their
talks at the AMS meeting at DePaul University.
- 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.
- 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.
- Thu 25 Oct: We do not meet at 11AM, but meet at 4PM for
the Algebraic Geometry Seminar of Frank Sottile.
- Thu 1 Nov: talk of Olga Kashcheyeva.
- Thu 8 Nov: We did not meet; attended the talks
of Joe Harris.
- Thu 15 Nov: presentation of Danko Adrovic
on tropical factorization.
- Thu 22 Nov: thanksgiving.
- Thu 29 Nov: Kathy Piret presents her talk for MACIS'07:
computing critical points by continuation.
Seminars organized in previous semesters: