Organisers : Benoît Bertrand, Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin

Max Planck Intitute, Bonn, July 13th 2012

Max-Planck-Institut für Mathematik

11:00 João Nunes (Instituto Superior Técnico, Lisboa)

Quantization and tropical geometry for toric manifolds

Let X be a toric manifold. We will describe families of Kahler structures on X which are interesting from the point of view of geometric quantization. Along each family, the Kahler metric collapses in the Gromov-Hausdorff sense to a Hessian metric on the moment polytope P. Moreover, as the Kahler structure degenerates, the amoebas for hypersurfaces in X tropicalize under an appropriate Legendre transformation of P. This is based on joint work with T.Baier, C.Florentino and J.Mourao.

14:00 Lothar Göttsche (ICTP, Trieste and MPI, Bonn)

Refined curve counting and refined Severi degrees.

This is report on joint works with Vivek Shende, Florian Block and Sam Payne. An old conjecture of mine gives a generating function for the numbers of $\delta$-nodal curves in linear systems on surfaces. In this talk we want to propose a refinement of the conjecture, where the numbers of curves are replaced by polynomials in a variable $y$, which for $y=1$ specialize to the numbers of curves. For rational surfaces these refined invariants are related to Welschinger invariants and have an interpretation in tropical geometry.

15:00 Tea

15:30 Sergey Galkin (Universität Wien)

Mutations of potentials and their upper bounds

We introduce Berenstein's notion of upper bounds to theory of mutations of potentials and prove an excessive Laurent phenomenon (which now says - upper bounds are preserved by mutations) using this new technique. The statement is quite general and can be applied in different contexts where Laurent polynomials and their mutations appear (e.g. Auroux's wall-crossing in symplectic geometry and mirror symmetry). This is a joint work with John Alexander Cruz Morales (preprint IPMU 12-0110).