Virtual School Geometry and Dynamics of Foliations

TALKS & DISCUSSION SESSIONS 
MAY 25 
MAY 26 
MAY 28 
14:00  14:40 
Brent Pym DISCUSSION SESSION 
Araujo  Druel DISCUSSION SESSION 

14:40  14:50 
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14:50  15:15 
Tiago Jardim da Fonseca 
Katsuhiko Okumura VIDEO 
Roberto Svaldi (40 min talk) 
15:15  15:20 
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15:20  15:45 
Federico Lo Bianco VIDEO 
YenAn Chen VIDEO 
 
15:45  15:50 
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15:50  16:15 
Rémi Jaoui VIDEO 
João Paulo Figueredo VIDEO 
Maxence Mayrand VIDEO 
16:15  16:20 
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16:20  17:00 
Frédéric Touzet DISCUSSION SESSION 
Cascini  Spicer DISCUSSION SESSION 
All the times above are local times of Marseille, France. ABSTRACTS & VIDEOS Bertrand Deroin  CNRS/IMPA, AGM Dynamics and topology of the Jouanolou foliation  VIDEO I will report on some joint work with Aurélien Alvarez, which shows that the Jouanolou foliation in degree two is structurally stable, and that it has a nontrivial domain of discontinuity. This result is opposed to a series of results beginning in the sixties with the works of HudaiVerenov and Ilyashenko. Tiago Jardim da Fonseca  University of Oxford Higher Ramanujan Foliations I will describe a remarkable family of higher dimensional foliations generalizing the equations studied by Darboux, Halphen, Ramanujan, and many others, and discuss some related geometric problems motivated by number theory. Federico Lo Bianco  Institut Camille Jordan (Lyon)  VIDEO On symmetries of transversely projective foliations. We consider a holomorphic (singular) foliation F on a projective manifold X and a group G of birational transformations of X which preserve F (i.e. it permutes the set of leaves). We say that the transverse action of G is finite if some finite index subgroup of G fixes each leaf of F. In a joint work with J. V. Pereira, E. Rousseau and F. Touzet, we show finiteness for the group of birational transformations of general type foliations with tame singularities and transverse finiteness for (nonvirtually euclidean) transversely projective foliations. In this talk I will focus on the latter result; time permitting, I will show how the presence of a transverse structure (projective, hyperbolic, spherical...) and the analysis of the resulting monodromy representation allow to reduce to the case of modular foliations on Shimura varieties and to conclude. Rémi Jaoui  University of Notre Dame  VIDEO A modeltheoretic analysis of geodesic equations in negative curvature. To any algebraic differential equation, one can associate a firstorder structure which encodes some of the properties of algebraic integrability and of algebraic independence of its solutions. To describe the structure associated to a given algebraic (nonlinear) differential equation (E), typical questions are:  Is it possible to express the general solutions of (E) from successive resolutions of linear differential equations?  Is it possible to express the general solutions of (E) from successive resolutions of algebraic differential equations of lower order than (E)?  Given distinct initial conditions for (E), under which conditions are the solutions associated to these initial conditions algebraically independent? In my talk, I will discuss in this setting one of the first examples of noncompletely integrable Hamiltonian systems: the geodesic motion on an algebraically presented compact Riemannian surface with negative curvature. I will explain a qualitative modeltheoretic description of the associated structure based on the global hyperbolic dynamical properties identified by Anosov in the 70’s for the geodesic motion in negative curvature. 
Katsuhiko Okumura – Waseda University Topics on the Poisson varieties of dimension at least four It is thought that the classifications and constructions of holomorphic Poisson structures are worth studying. The classification when the Picard rank is 2 or higher is unknown. In this talk, we will introduce the classification of holomorphic Poisson structures with the reduced degeneracy divisor that have only simple normal crossing singularities, on the product of Fano variety of Picard rank 1. This claims that such a Poisson manifold must be a diagonal Poisson structure on the product of projective spaces, so this is a generalization of Lima and Pereira's study. The talk will also include various examples, classifications, and problems of highdimensional holomorphic Poisson structures. YenAn Chen University of Utah  VIDEO Boundedness of Minimal Partial du Val Resolutions of Canonical Surface Foliations. By the work of Brunella and McQuillan, it is known that smooth foliated surfaces of general type with only canonical singularities admit a unique canonical model. It is then natural to wonder if these canonical models have a good moduli theory and, in particular, if they admit a moduli functor. In this talk, I will show that the canonical models and their minimal partial du Val resolutions are bounded. João Paulo Figueredo  IMPA VIDEO Regular foliations on rationally connected manifolds. In this talk, we will consider the problem of classifying regular foliations on rationally connected manifolds over the complex numbers. Conjecturally, these foliations should have algebraic leaves. I will show this is true when the manifold has dimension three, and the foliation has codimension one and non pseudoeffective canonical bundle. Roberto Svaldi  EPFL Foliations and MMP: some applications I will explain how the recent developments in the theory of birational geometry of rank 2 foliations on 3folds have find quite a few applications in the study of the structure of such foliations and their singularities. In this talk that is a complement to the course of Cascini and Spicer, but which will be selfcontained, I shall briefly explain why the MMP terminates and then proceed to illustrate a few applications of these techniques, e.g., to the classification of canonical singularities, to the study of adjunction theory, and to the study of hyperbolicity properties of foliated 3folds. The work is in collaboration with Calum Spicer. Maxence Mayrand – University of Toronto  VIDEO Hyperkähler structures on symplectic realizations of holomorphic Poisson surfaces I will discuss the existence of hyperkähler structures on local symplectic groupoids integrating holomorphic Poisson manifolds, and show that they always exist when the base is a Poisson surface. The hyperkähler structure is obtained by constructing the twistor space by lifting specific deformations of the Poisson surface adapted from Hitchin's unobstructedness result. In the special case of the zero Poisson structure, we recover the FeixKaledin hyperkähler structure on the cotangent bundle of a Kähler manifold. 