Conference on Complex Geometry in Mérida – Alberto 80

Bertrand Deroin, Cergy Universite: The Dichotomy Structural Stability/Bifurcations in Moduli Spaces of Holomorphic Foliations on Surfaces.

In these lectures, I will report on: the history of the problem, Hudai-Verenov and Ilyashenko's rigidity theorems and their descendants, the case of Riccati equations and their relations with Kleinian groups, and finally discuss some new examples of non linear structurally stable foliations on the complex projective plane.

Andrés Navas, Universidad de Santiago de Chile: Arc-Connectedness of the Space of 1-Dimensional Commuting Diffeomorphisms

Most of this minicourse will be devoted to explain the ideas of a recent work in collaboration with Hélène Eynard-Bontemps (Inst. Fourier, Grenoble) about connectedness properties of the space of commuting diffeomorphisms of 1-manifolds. Many questions will be addressed along the path, some of them related to the complex(analytic) case.

Juan Manuel Burgos, CINVESTAV

Angel Cano, UNAM: Knot groups Representations into Matrix Groups via Gram-Schmidt

In this talk we propose a computacional method to calculate matrix representations of Knot fundamental groups. Joint work with Ángel Rodríguez a Patricia Domínguez.

Gabriela Hinojosa, UAEM: Hausdorff Dimension of Equivalent Dynamically Defined Wild Knots

Let T be a pearl necklace consisting of the union of n consecutive tangent closed 3-balls Bi (i=1,2,..., n) and consider the Kleinian group ΓT generated by the reflections on the boundaries ∂Bi. Let Λ(ΓT) be a wild knot obtained as the limit set of ΓT acting on the 3-sphere S3. We say that a n-pearl necklace V consisting of the union of consecutive tangent closed 3-balls Ci (i=1,2,..., n) is equivalent to T if there exists a homeomorphism φ:S3→S3 such that φ (V)=T, φ(Ci)=Bi, and φ (Ci∩Ci+1)=Bi∩Bi+1. In this talk we will study the Hausdorff dimension of Λ(ΓT) for equivalent pearl necklaces.

Laurent Meersseman, Universidad de Angers: Moduli Stacks in Complex Analytic Geometry

We explain how stack theory can be used in Complex Analytic Geometry to go beyond the classical local deformation theory of Kodaira-Spencer and Kuranishi and to deal with global moduli problems. After recalling the basic notions of families, deformations and the main theorems on the existence of a semi-universal deformation, we introduce the Teichmu¨ller and moduli stacks as a natural setting to globalize the questions. Then we go back to a local point of view and study the local structure of the Teichmu¨ller stack, especially in the K¨ahler case

Bertrand Deroin, Cergy Universite: The Dichotomy Structural Stability/Bifurcations in Moduli Spaces of Holomorphic Foliations on Surfaces.

In these lectures, I will report on: the history of the problem, Hudai-Verenov and Ilyashenko's rigidity theorems and their descendants, the case of Riccati equations and their relations with Kleinian groups, and finally discuss some new examples of non linear structurally stable foliations on the complex projective plane.

Eleonora Di Nezza Sorbonne Universite: Families of Kahler-Einstein Metrics

We study families of singular K¨ahler-Einstein metrics. We develop the first steps of pluripotential theory in family in order to obtain an explicit control of the C⁰ estimate even if the complex structure changes. We can then treat several geometric context such as families of general type vari- eties and families of Calabi-Yau varieties: We show that the potential of a (singular) K¨ahler-Einstein metric on the generic fibre converges to the one in the central fibre.

Hugo García Compean, CINVESTAV: A Perturbative Approach to Average Asymptotic Invariants for Knots and Links

The perturbative expansion of Chern–Simons gauge theory leads to invariants of knots and links, the so-called finite type invariants or Vassiliev invariants. It has been proved that at any order in perturbation theory the superposition of certain amplitudes is an invariant of that order. Bott–Taubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological framework. One of the consequences of this formalism is that the resulting amplitudes are rewritten in cohomological terms in configuration spaces. This cohomological structure can be used to translate Bott–Taubes integrals into Chern–Simons perturbative amplitudes and vice versa. In this talk, this program is performed up to third order in the coupling constant. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the 3-manifold. The Bott–Taubes integrals obtained are used for constructing higher-order average asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.

Kyoung Seog Lee, University of Miami IMSA

Andrés Navas, Universidad de Santiago de Chile: Arc-Connectedness of the Space of 1-Dimensional Commuting Diffeomorphisms

Most of this minicourse will be devoted to explain the ideas of a recent work in collaboration with Hélène Eynard-Bontemps (Inst. Fourier, Grenoble) about connectedness properties of the space of commuting diffeomorphisms of 1-manifolds. Many questions will be addressed along the path, some of them related to the complex(analytic) case.

Laurent Meersseman, Universidad de Angers: Moduli Stacks in Complex Analytic Geometry

We explain how stack theory can be used in Complex Analytic Geometry to go beyond the classical local deformation theory of Kodaira-Spencer and Kuranishi and to deal with global moduli problems. After recalling the basic notions of families, deformations and the main theorems on the existence of a semi-universal deformation, we introduce the Teichmu¨ller and moduli stacks as a natural setting to globalize the questions. Then we go back to a local point of view and study the local structure of the Teichmu¨ller stack, especially in the K¨ahler case

Sorin Dumitrescu, Universite Cote d’Azur: Holomorphic sl(2, C)– differential systems on compact Riemann surfaces and curves in compact quotients of SL(2, C)

We explain the strategy of a recent result that constructs holomorphic sl(2, C)–differential systems over some Riemann surfaces Σ_g of genus g ≥ 2, such that the image of the associated monodromy homomorphism is some cocompact Kleinian subgroup Γ ⊂ SL(2, C). As a consequence, there exist holomorphic maps from Σ_g to the quotient SL(2, C)/Γ, that do not factor through any elliptic curve. This answers positively a question asked by Huckleberry and Winkelmann, also raised by Ghys. This is a joint work with Indranil Biswas (TIFR, Mumbai), Lynn Heller (BIMSA, Beijing) and Sebastian Heller (BIMSA, Beijing).

Bertrand Deroin, Cergy Universite: The Dichotomy Structural Stability/Bifurcations in Moduli Spaces of Holomorphic Foliations on Surfaces.

In these lectures, I will report on: the history of the problem, Hudai-Verenov and Ilyashenko's rigidity theorems and their descendants, the case of Riccati equations and their relations with Kleinian groups, and finally discuss some new examples of non linear structurally stable foliations on the complex projective plane.

Laurent Meersseman, Universidad de Angers: Moduli Stacks in Complex Analytic Geometry

We explain how stack theory can be used in Complex Analytic Geometry to go beyond the classical local deformation theory of Kodaira-Spencer and Kuranishi and to deal with global moduli problems. After recalling the basic notions of families, deformations and the main theorems on the existence of a semi-universal deformation, we introduce the Teichmu¨ller and moduli stacks as a natural setting to globalize the questions. Then we go back to a local point of view and study the local structure of the Teichmu¨ller stack, especially in the K¨ahler case

Erwan Rousseau, Universite de Brest: Numerically nonspecial varieties

Campana introduced the class of special varieties as the varieties admit- ting no Bogomolov sheaves i.e. rank one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Those are precisely varieties which do not admit any surjective map onto a general type orbifold. Campana raised the question if one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank one coherent subsheaf of the cotangent bundle with numerical dimension 1 is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the function field setting (joint work with Frederic Touzet and Jorge Pereira).

Santiago López de Medrano, Imate, UNAM: On Singularities of Intersections of Concentric Ellipsoids in Rⁿ

We will briefly recall what is known about the topology of intersections of concentric ellipsoids in $\R^n$, starting with the intersections of two of them. We will present some new singular examples and their smoothings. In some cases we will also consider the corresponding associated moment-angle manifolds that admit complex structures in the smooth case. This is joint work with Enrique Artal and María Teresa Lozano from the University of Zaragoza, Spain.

Aubin Arroyo, UNAM Cuernavaca

9:30am

Andres Navas, Universidad de Santiago de Chile: Arc-Connectedness of the Space of 1-Dimensional Commuting Diffeomorphisms

Most of this minicourse will be devoted to explain the ideas of a recent work in collaboration with Hélène Eynard-Bontemps (Inst. Fourier, Grenoble) about connectedness properties of the space of commuting diffeomorphisms of 1-manifolds. Many questions will be addressed along the path, some of them related to the complex(analytic) case.

Ludmil Katzarkov, University of Miami IMSA

Adolfo Guillot, UNAM Mexico