3. Postdoctoral Program - MSRI

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While at MSRI, Gathmann extended the construction of tropical moduli spaces to cases of curves
of higher genus or whose ambient spaces are more general tropical varieties than vector spaces.
9. Ilia Itenberg, Viatcheslav Kharlamov and Eugenii Shustin worked on a paper devoted to
Welschinger invariants of small Del Pezzo surfaces. It contains a recursive formula for purely
real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown
up at q real and s ≤ 1 pairs of conjugate imaginary points, where q + 2s ≤ 5, and the real
quadric blown up at s ≤ 1 pairs of conjugate imaginary points and having non-empty real
part. The formula is similar to Vakil’s recursive formula for Gromov-Witten invariants of these
surfaces and generalizes the recursive formula for purely real Welschinger invariants of real
toric Del Pezzo surfaces (the latter formula was obtained earlier by Itenberg, Kharlamov and
Shustin). The consequences of the formula include the positivity of the Welschinger invariants
under consideration and their logarithmic asymptotic equivalence to genus zero Gromov-Witten
invariants.
10. Ilia Itenberg, Grigory Mikhalkin and Ilia Zharkov have advanced on a project on tropical
homology. The outcome of this project is a definition of homology (and cohomology) groups
enhanced with the tropical Picard-Fuchs operator responsible for Schmid’s Mixed Hodge Structure
of the degeneration. These tropical objects are expected to match with the corresponding
classical objects in the case when the tropical manifold (smooth in the coarse sense) comes as a
tropical limit of a 1-parametric family of complex manifolds. Furthermore the resulting framework
allows to dualize the tropical set-up (according to the Mirror Symmetry principles) to get a
different type of homology groups conjecturally responsible for deformations. A particularly attractive
feature is similarity of the geometric object corresponding to the Picard-Fuchs operator
(tropical wave) with the tropical hyperplane section.
11. Eric Katz and Sam Payne finished an article on realization spaces for tropical fans. They
introduced a moduli functor for varieties whose tropicalization realizes a given weighted fan and
showed that this functor is an algebraic space in general, and is represented by a scheme of finite
type when the associated toric variety is quasiprojective. They also studied the geometry of
tropical realization spaces for the matroid fans studied by Ardila and Klivans, and show that
the tropical realization space of a matroid fan is a torus torsor over the realization space of the
matroid. One consequence is that these tropical realization spaces satisfy Murphy’s Law.
12. Ludmil Katzarkov, Grigory Mikhalkin and Ilia Zharkov worked on tropical Jacobians in
higher dimension (in particular, intermediate Jacobians) and Prymians. The outcome of this
ongoing project was a refinement of the intermediate Jacobian in the tropical case and connection
between this refinement and the tropical wave corresponding to the Picard-Fuchs operator.
13. Diane Maclagan worked on the tropical inverse problem. Here the starting point is the fact
that every tropical curve (meaning a weighted balanced one-dimensional rational polyhedral
complex) is the tropicalization of a curve in the torus, and the tropical inverse problem asks for
which tropical complexes this is the case. She studied various variants of this question, and she
explored its relevance to birational geometry.
14. Hannah Markwig, Thomas Markwig and Eugenii Shustin completed a paper, titled Tropical
curves with a singularity in a fixed point, which concerns families of curves with a singularity
in a fixed point. The tropicalization of such a family is a linear tropical variety. They describe
its maximal dimensional cones using results on linear tropical varieties due to Ardila–Klivans
and Feichtner–Sturmfels. They show that a singularity tropicalises either to a vertex of higher
valence or of higher multiplicity, or to an edge of higher weight. They also classify maximal
dimensional types of singular tropical curves. For those, the singularity is either a crossing of
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