Title: Metrics of constant curvature on Algebraic Manifolds

Metrics of constant curvature have always been thought as the best riemannian metrics we can equip a given manifold with. In the first part of the talk I will explain few classical reasons supporting this belief and  some of the subtleties we face proving their existence.

In particular real and complex manifolds behave very differently depending on which curvature (Sectional, Ricci, scalar…) one tries to make constant.

In the second part of the talk I will focus on the complex case, introducing some fundamental obstructions and analysing if and when few basic geometric operations such as Galois coverings, blow ups and resolutions of singularities preserve the existence of such metrics.

A number of open problems will be discussed.

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Title: Ergodic theorems for laminations and foliations: recent results and perspectives

This report discusses recent results as well as new perspectives in the ergodic theory for hyperbolic Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are directed positive harmonic currents, multiplicative cocycles and leafwise Poincar ́e metric. We deal with various ergodic theorems for such laminations: Random Ergodic Theorem, Geometric Birkhoff Ergodic Theorem, Oseledec Multiplicative Ergodic Theorem and Unique Ergodicity Theorems. Applications of these theorems are also given. In particular, we define and study the canonical Lyapunov exponents for a large family of singular holomorphic foliations on compact projective surfaces. 

Topological and algebro-geometric interpretations of these characteristic numbers are also treated. These results highlight the strong similarity as well as the fundamental differences between the ergodic theory of maps and that of Riemann surface laminations.

Keywords: Riemann surface lamination, singular holomorphic foliation, positive har-monic currents, multiplicative cocycles, leafwise Poincar ́e metric, ergodic theorems, Lya-punov exponents.

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