3. Postdoctoral Program - MSRI

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6. Alex Fink was an invited speaker at the research workshop in October, where he presented his
project on tropical cycles and Chow polytopes. He associates a Chow polytope to any abstract
tropical variety in R n , using a Minkowski sum operation on tropical varieties. This construction
generalizes several previously known associations of polyhedra to certain tropical varieties.
7. Benjamin Iriarte wrote a paper on Phylogenetic trees and the tropical Grassmannian. One of
the main problems in evolutionary biology is that of reconstructing a phylogenetic tree from a
DNA sequence alignment of n species. This process is considerably simplified by the distance
based approach. In order to really make this approach fruitful, one needs to understand dissimilarity
vectors of trees, which leads to the still unsolved problem of characterizing generalized
m-dissimilarity vectors of n-trees. It turns out there is a natural relation between these vectors
and the corresponding tropical Grassmannian Gm,n, and this opens the door to tropical geometry
as a possible tool to solve the characterization problem. Iriarte identifies the precise relation
between these two sets. This resolves a problem stated by Pachter and Speyer in 2003.
8. Kirsten Schmitz, jointly with her advisor Tim Römer, developed a theory of Generic tropical
varieties. She showed that in the constant coefficient case the generic tropical variety of a
graded ideal exists. This can be seen as the analogon to the existence of the generic initial ideal
in Groebner basis theory. She also determined the generic tropical variety as a set in general
and as a fan for principal ideals, linear ideals and ideals in low dimension.
9. The fruitful interaction with the Symplectic Geometry program was highlighted by the intriguing
and geometrically beautiful research project of Nick Sheridan, which relates coamoebas
of the hyperplanes with the Fukaya category of the projective planes. Apparently both these
things can be described (in some sense) by a single immersed Lagrangian sphere.
10. Cynthia Vinzant wrote a paper on Real radical initial ideals. In this work she explored the
consequences of an ideal I of real polynomials having a real radical initial ideal, both for the
geometry of the real variety of I and as an application to sums of squares representations of
polynomials. She showed that if inw(I) is real radical for a vector w in the tropical variety, then
w is in the logarithmic set of the real variety. We also give algebraic sufficient conditions for w
to be in the logarithmic limit set of a more general semialgebraic set. If in addition the entries
of w are positive, then the corresponding quadratic module is stable. In particular, if inw(I)
is real radical for a positive vector w then the set of sums of squares modulo I is stable. This
provides a method for checking the conditions for stability given by Powers and Scheiderer.
The community of graduate students and postdocs formed at MSRI will continue to interact
fruitfully in the coming years. One early indication for this was the one-day workshop on Tropical
Geometry at TU Berlin in December 2009 which was organized by doctoral students who had
only met a few months earlier, at MSRI during the Introductory Workshop in August 2009.
9. Nuggets and Breakthroughs
A particularly exciting development was the emerging connection between tropical geometry
and number theory, which was highlighted by the work of Matt Baker, Vladimir Berkovich,
Walter Gubler and Sam Payne. This was enabled by Payne’s remarkable result that Berkovich’s
analytification of an algebraic variety is the inverse limit of all tropical varieties obtained by
choosing a concrete embedding. Walter Gubler solved the longstanding Bogomolov conjecture on
equidistribution of points of bounded height on abelian varieties using tropical analytic geometry.
The breakthroughs made during the program and concerning connections between tropical,
complex and symplectic geometries are the following ones.
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