3. Postdoctoral Program - MSRI

6. Research Developments
Here is a sample of the many new research developments that emerged during the program:
1. Mohammed Abouzaid, Mark Gross and Bernd Siebert have started their work on the tropical
version of the Fukaya category. This project mends tropical curve and tropical-looking gradient
flows that appeared in the work of Fukaya and Oh. The outcome allows to compute not only
the number of holomorphic disks, but to incorporate the information about their area which is
crucial for computing the superpotential.
2. Vladimir Berkovich worked on the development of analytic geometry over the field of one
element. This provides a systematic new approach to the analytification of algebraic varieties,
and its connection to tropical geometry.
3. Erwan Brugallé and Grigory Mikhalkin have worked on realizability of superabundant tropical
curves extending the realizability criterion found by David Speyer for elliptic curves. They have
used tropical modification to extend Speyer’s criterion to arbitrary genus case.
4. Erwan Brugallé and Lucia Lopez de Medrano have used tropical modifications to locate (and
relate to the real geometry case) tropical singularities, in particular tropical inflection points.
5. Dustin Cartwright, Mathias Häbich, Bernd Sturmfels and Annette Werner started to write
a joint paper on Mustafin varieties. Such a variety is a degeneration of projective space induced
by a point configuration in a Bruhat-Tits building. The special fiber is reduced and
Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For
configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled
simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision.
This is the connection to tropical and toric geometry. For general configurations, the irreducible
components of the special fiber are rational varieties, and any blow-up of projective space along
a linear subspace arrangement can arise. A detailed study of Mustafin varieties was undertaken
for configurations in the Bruhat-Tits tree of P GL(2) and for triangles in the building of P GL(3).
6. Jan Draisma has begun to develop a theory of tropical reparameterizations. Given a classical
polynomial map f : A m → A n between affine spaces parameterizing a variety X = im(f), the
aim is to construct a coordinate change a : A p → A n such that the composition f ◦ a tropicalizes
naively (that is, by replacing + by min and × by +) to a tropical polynomial map whose image is
all of trop(X). This has the appealing interpretation that trop(X) can be ”folded” from a piece
of p-dimensional paper. At MSRI, Draisma systematically classified known examples where this
is the case, and he proved the existence of coordinate changes that are good locally.
7. Anton Dochtermann, Michael Joswig and Raman Sanyal completed a paper on tropical types
and associated cellular resolutions. An arrangement of tropical hyperplanes in the tropical torus
leads to a notion of ‘type’ data for points, with the underlying unlabeled arrangement giving
rise to ‘coarse type’. The decomposition of the tropical torus induced by types gives rise to
minimal cocellular resolutions of certain associated monomial ideals. Via the Cayley trick from
geometric combinatorics this also yields cellular resolutions supported on mixed subdivisions of
dilated simplices, extending previously known constructions. Moreover, the methods developed
lead to an algebraic algorithm for computing the facial structure of arbitrary tropical complexes.
8. Andreas Gathmann and his students from Kaiserslautern continued their research on tropical
enumerative geometry. In algebraic geometry, enumerative problems are usually studied by
constructing moduli spaces of curves or stable maps to some variety. In the tropical world the
corresponding spaces have only been constructed so far for rational curves in a real vector space.
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