__Title pretalk__: Foliations and birational geometry.

__Abstract pretalk__: In this talk I will introduce the notion of foliation on an algebraic variety.

I will focus on the basic notions involved in the study of the birational geometry of foliations, and will give some examples and motivations for the study of such objects.

No prior knowledge/familiarity with the concept of foliation will be assumed.

__Title talk:__

Towards a moduli theory for canonical models of foliated surfaces of general type.

__Abstract talk__:

In recent years there has been considerable progress in extending the ideas and techniques of the Minimal Model Program beyond the realm of algebraic varieties to the study of foliations. For the case of foliations on surfaces, McQuillan, Brunella and Mendes have obtained a detailed classification — analogous to the Enriques-Kodaira classification.

In this seminar, I will explain how, using the birational classification of foliations on surfaces and MMP techniques, we can start constructing moduli spaces for minimal foliations that have maximum Kodaira dimension on surfaces. While there are many similarities between the birational variational theory and the theory of foliations, a whole cornucopia of new phenomena appears in the latter. Accordingly, I will try to explain:

(1) what these new phenomena are;

(2) what new difficulties they introduce into the identification of a good functor of moduli for the aforementioned foliations;

(3) how these difficulties can be overcome to obtain a satisfactory theory of moduli and a construction of the associated moduli space for foliated surfaces of general type.

The talk will feature joint work with C. Spicer, and joint work in progress with M. McQuillan, C. Spicer, and S. Velazquez.