Componentes tipo pullback do espaço de folhações de codimensão um em \mathbb{P}^n

Speaker: Viviana Ferrer, IME-UFF.

Date: 29 may 2019, 14h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: O espaço de folheações holomorfas de codimensão um em \mathbb{P}^n tem uma componente irredutível cujo elemento genérico pode ser escrito como o pullback F^*\mathcal{F}, onde \mathcal{F} é uma folheação genérica de \mathbb{P}^2 e F :\mathbb{P}^n\dasharrow \mathbb{P}^2 é um mapa racional. (Cerveau, Lins-Neto, Edixhoven, 2001).

Nesta palestra mostraremos como são encontradas fórmulas para o grau desta componente no caso de pullback linear.
Mostraremos também quais são as dificuldades para parametrizar (e consequentemente calcular o grau) componentes dadas por pullback de folheações por mapas não lineares.

 

Lyapunov exponents of the Brownian motion on a Kähler manifold

Speaker: Bertrand Deroin, CNRS-AGM, Cergy.

Date: 17 apr 2019, 14h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: If E is a flat bundle of rank r over a Kähler manifold X, we define the Lyapunov spectrum of E: a set of r numbers controlling the growth of flat sections of E, along Brownian trajectories. We show how to compute these numbers, by using harmonic measures on the foliated space P(E). Then, in the case where X is compact, we prove a general inequality relating the Lyapunov exponents and the degrees of holomorphic subbundles of E and we discuss the equality case.

This is based on joint work with J. Daniel.

 

Algebraic solutions of Irregular Garnier systems

Speaker: Frank Loray, CNRS-IRMAR, Rennes.

Date: 27 mar 2019, 14h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: We prove that algebraic solutions of Garnier systems in the irregular case are of two types, generalizing a result of Ohyama and Okumura for Painlevé equations (rank N=1). The so called "classical solutions" come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank N = 2 case (two indeterminates). The "pull-back solutions" come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide a complete list when the differential Galois group is SL2(C).

By the way, we have a complete list of algebraic solutions for the rank N = 2 irregular Garnier systems. The rank N=1 case correspond to Painlevé equations I to V and is classical; it has been revisited from this point of view by Ohyama and Okumura.

This is joint work with Karamoko Diarra (Bamako University, Mali).

 

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