Programme

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

Toulouse, 4 October 2012

Institut de mathématiques de Toulouse, Amphi Schwartz Bât. 1R3

10h:30 Coffee

11h Rémi Crétois (Université de Genève)

Real automorphisms of a vector bundle and the determinant of Cauchy-Riemann operators

We consider a complex vector bundle N equipped with a real structure cN over an abstract real curve. The set of all Cauchy-Riemann operators on (N,cN) is a contractible space. The determinant bundle over this space is a real line bundle whose fiber at a given operator is its determinant. We will try to describe the action of the automorphism group of (N,cN) on the orientations of the determinant bundle and give some consequences on the first Stiefel-Whitney class of the moduli spaces of real pseudo-holomorphic curves in a real symplectic manifold.

14:00 Martijn Kool (Imperial College)

Reduced classes and curve counting on surfaces

Counting nodal curves in linear systems |L| on smooth projective surfaces S is a problem with a long history. The Göttsche conjecture, now proved by several people, states that these counts are universal and only depend on c1(L)2, c1(L).c1(S), c1(S)2 and c2(S). We present a quite general definition of reduced Gromov-Witten and stable pair invariants on S. The reduced stable pair theory is entirely computable. Moreover, we prove that certain reduced Gromov-Witten and stable pair invariants with many point insertions coincide and are both equal to the nodal curve counts appearing in the Göttsche conjecure. This can be see as version of the MNOP conjecture for the canonical bundle KS. This is joint work with R. P. Thomas.

15h coffee

15h30 Vladimir Fock (Université de Strasbourg)

Integrable systems, dimers and plane curves.

We present a construction of a class of integrable systems : starting from a Newton polygon we (following Goncharov and Kenyon) construct a cluster variety and a Poisson map to the space of plane curves with the given Newton polygon. The fibres of the map are Lagrangian subvarieties. We show that the same systems can be obtained from affine Lie groups. We will discuss the geometry and the combinatorics of the construction.