Speaker: Christopher D. HACON (Utah)

Title: Classification of Algebraic Varieties I and II

Abstract: A complex projective variety is a subset of complex projective space defined by a set of homogeneous polynomials. In this talk I will discuss recent results that describe the geometry of these varieties. The case of varieties of dimension 1, also known as Riemann surfaces, is classical. The geometry of smooth varieties of dimension 2 was understood by the Italian school of Algebraic Geometry at the beginning of the 20-th century. The Minimal Model Program is an attempt to generalize these results to higher dimension. The 3 dimensional case was understood in the 1980's by celebrated work of Mori and others. In these talks I will begin by reviewing the classification of algebraic varieties in low dimension and then I will discuss joint work with Birkar, Cascini and McKernan towards extending the Minimal Model Program to arbitrary dimension. In particular I will discuss the following:

Theorem: The canonical ring of any smooth projective algebraic variety is finitely generated. (Note that this Theorem was independently proven by Y.-T. Siu.)