Holomorphic foliations in Alsace

Conference posters

The poster sessions will take place during the coffee breaks on Tuesday afternoon, Wednesday and Thursday

• Jinan LOUBANI (Toulouse, France)
Moduli spaces of a family of topologically non quasi-homogeneous functions.
We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi-homogeneous. Each element $f$ in this class induces a germ of foliation $(df = 0)$. This poster presents a local result about the moduli spaces of the foliations in this class. More precisely, it gives the dimension of the tangent space to the moduli space, describes the local moduli space and gives local analytic normal forms. It also presents a result regarding the uniqueness of these normal forms.

Newton non-degenerate foliations
We introduce the concept of Newton non-degenerate foliations, generalizing the well-known definition of Oka (1996) for germs of plane curves. We prove that this foliations are exactly the ones that have a combinatorial reduction of singularities, in particular it is given by a reduction of singularities of the associated Newton polyhedra system.

• Younes NIKDELAN (Rio de Janeiro, Brazil)
Modular vector fields attached to Dwork family
This poster is based on my latest work with H. Movasati (see arXiv:1603.09411), which is a part of the project called Gauss-Manin Connection in Disguise, started by H. Movasati. Here we introduce a moduli space $\sf{T}$ of the enhanced Calabi-Yau $n$-folds arising from Dwork family and describe a unique vector field $\sf{R}$, calling modular vector field, in $\sf{T}$, whose stisfies certain properties with respect to the underlying Gauss-Manin connection. For $n=1,2$ we compute explicit expressions of $\sf{R}$ and give a solution of $\sf{R}$ in terms of quasi-modular forms. Somehow, we can consider $\sf R$ as a generalization of Darboux-Halphen and Ramanujan systems of ordinary differential equations.

• Valente RAMíREZ (Rennes, France)
An example of a non-algebraizable singularity
The existence of non-algebraizable singularities was discovered by Genzmer and Teyssier in 2010, where they prove the existence of countably many classes of saddle-node singularities which are not algebraizable. Their proof, however, is of an existential nature and does not provide any concrete examples. In this poster we will present the first explicit example of a non-algebraizable singularity. The construction is based on the formal classification of non-dicritic degenerate singularities given by Ortiz-Bobadilla, Rosales-González and Voronin 2012. This example was constructed in collaboration with Frank Loray.

• Maria elenice RODRIGUES HERNANDES (Maringá, Brazil)
Parametrized Monomial Surfaces in $\mathbb{C}^4$
Monomial maps are an interesting class that was explored by many authors. In this work we classify germs of parametrized monomial surfaces defined by $f:(\mathbb{C}^2,0) \to (\mathbb{C}^4,0)$ that are ${\mathcal A}$-finitely determined, where ${\mathcal A}$ is the group of right-left equivalences. In particular we can associated to $f(0,y)$ a parametrized plane curve whose semigroup of values is the main tool to the classification of parametrized monomial surfaces, and important to obtain some formulae or estimate of invariants associated to this class of map germs. This is a joint work with Maria Aparecida Soares Ruas.

• Yadollah ZARE (Rio de janiero, Brazil)
Pull-back foliations and center problem
The space of polynomial differential equations of a fixed degree and with a center singularity has many irreducible components. We prove that pull-back differential equations form an irreducible component of such an space. The main tools are Picard-Lefschetz theory of a polynomial with complex coefficients in two variables, Dynkin diagram of the polynomial , iterated integral , Brieskorn module.