|Location:||MSRI: Baker Board Room|
Oliver Fabert: "Transversality for J-holomorphic curves"
J-holomorphic curves are the most important tools to study the global properties of symplectic manifolds and also play an important role in string theory. In order to define the desired invariants (observables), the set of J-holomorphic curves has to be equipped with a smooth structure of dimension expected by a topological index. For this it suffices to prove that the Cauchy-Riemann operator, viewed as a section in an infinite-dimensional bundle, is transversal to the zero section in this
bundle. In my talk I will review methods to achieve the desired transversality result and illustrate in particular how this leads to a
difference between J-holomorphic curves and honest holomorphic curves (one-dimensional complex submanifolds) in the case when the symplectic manifold is Kahler, i.e., also carries a (compatible) complex structure.
Mounir Nisse: "Complex and non-Archimedean coamoebas"
Amoebas (resp. Coamoebas) are the link between the classical complex geometry and the tropical (resp. complex tropical) geometry. I will start by briefly introducing these objects in the complex algebraic hypersurfaces cases. The purpose of my talk will be to explain the relation between complex and non-Archimedean coamoebas on one hand, and Newton polytope on
the other hand. Moreover, a brief survey of the further development ofcomplex and non-Archimedean amoebas will be given, as well as a description of some new results. However, the same circle of ideas used on amoebas, also shows that the coamoebas have a similar geometric and combinatorial structure. Part of this work is in preparation jointly with M. Passare in the
complex case, and with F. Sottile in the non-Archimedean case. Application for n=2 will be outlined. Many examples, with pictures, will be given.